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| references
list | answers
dict |
|---|---|---|---|---|---|---|---|
gem-squad_v2-train-111000
|
572823f82ca10214002d9ec8
|
Group_(mathematics)
|
Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations
|
What is a finite composition of two symmetries or their inverses?
|
What is a finite composition of two symmetries or their inverses?
|
[
"What is a finite composition of two symmetries or their inverses?"
] |
{
"text": [
"every symmetry of the square"
],
"answer_start": [
403
]
}
|
gem-squad_v2-train-111001
|
5a81cbef31013a001a334ee5
|
Group_(mathematics)
|
Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations
|
What do quotient groups describe by themselves?
|
What do quotient groups describe by themselves?
|
[
"What do quotient groups describe by themselves?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111002
|
5a81cbef31013a001a334ee6
|
Group_(mathematics)
|
Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations
|
What presentation describes every group?
|
What presentation describes every group?
|
[
"What presentation describes every group?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111003
|
5a81cbef31013a001a334ee7
|
Group_(mathematics)
|
Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations
|
How many elements does the dihedral group generate?
|
How many elements does the dihedral group generate?
|
[
"How many elements does the dihedral group generate?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111004
|
5a81cbef31013a001a334ee8
|
Group_(mathematics)
|
Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations
|
What of a square is an infinite composition?
|
What of a square is an infinite composition?
|
[
"What of a square is an infinite composition?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111005
|
5a81cbef31013a001a334ee9
|
Group_(mathematics)
|
Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations
|
Which two elements does D4 generate?
|
Which two elements does D4 generate?
|
[
"Which two elements does D4 generate?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111006
|
5728250a4b864d1900164588
|
Group_(mathematics)
|
Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map H → G, i.e. any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map G → G / N.y[›] Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In general, homomorphisms are neither injective nor surjective. Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon.
|
What map shows the relation between sub and quotient groups?
|
What map shows the relation between sub and quotient groups?
|
[
"What map shows the relation between sub and quotient groups?"
] |
{
"text": [
"injective map"
],
"answer_start": [
93
]
}
|
gem-squad_v2-train-111007
|
5728250a4b864d1900164589
|
Group_(mathematics)
|
Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map H → G, i.e. any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map G → G / N.y[›] Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In general, homomorphisms are neither injective nor surjective. Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon.
|
What are the opposites of injective maps?
|
What are the opposites of injective maps?
|
[
"What are the opposites of injective maps?"
] |
{
"text": [
"surjective maps"
],
"answer_start": [
224
]
}
|
gem-squad_v2-train-111008
|
5728250a4b864d190016458a
|
Group_(mathematics)
|
Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map H → G, i.e. any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map G → G / N.y[›] Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In general, homomorphisms are neither injective nor surjective. Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon.
|
What is an example of a surjective map?
|
What is an example of a surjective map?
|
[
"What is an example of a surjective map?"
] |
{
"text": [
"canonical map"
],
"answer_start": [
298
]
}
|
gem-squad_v2-train-111009
|
5728250a4b864d190016458b
|
Group_(mathematics)
|
Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map H → G, i.e. any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map G → G / N.y[›] Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In general, homomorphisms are neither injective nor surjective. Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon.
|
What theory address the phenomenon of homomorphisms being neither injective nor surjective?
|
What theory address the phenomenon of homomorphisms being neither injective nor surjective?
|
[
"What theory address the phenomenon of homomorphisms being neither injective nor surjective?"
] |
{
"text": [
"the first isomorphism theorem"
],
"answer_start": [
599
]
}
|
gem-squad_v2-train-111010
|
5a81cd1f31013a001a334eef
|
Group_(mathematics)
|
Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map H → G, i.e. any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map G → G / N.y[›] Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In general, homomorphisms are neither injective nor surjective. Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon.
|
Subset G of H is seen as what map?
|
Subset G of H is seen as what map?
|
[
"Subset G of H is seen as what map?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111011
|
5a81cd1f31013a001a334ef0
|
Group_(mathematics)
|
Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map H → G, i.e. any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map G → G / N.y[›] Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In general, homomorphisms are neither injective nor surjective. Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon.
|
What is the minimum number of elements in a target?
|
What is the minimum number of elements in a target?
|
[
"What is the minimum number of elements in a target?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111012
|
5a81cd1f31013a001a334ef1
|
Group_(mathematics)
|
Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map H → G, i.e. any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map G → G / N.y[›] Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In general, homomorphisms are neither injective nor surjective. Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon.
|
What kind of map is similar to an injective map?
|
What kind of map is similar to an injective map?
|
[
"What kind of map is similar to an injective map?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111013
|
5a81cd1f31013a001a334ef2
|
Group_(mathematics)
|
Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map H → G, i.e. any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map G → G / N.y[›] Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In general, homomorphisms are neither injective nor surjective. Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon.
|
What theory discusses the surjective nature of canonical maps?
|
What theory discusses the surjective nature of canonical maps?
|
[
"What theory discusses the surjective nature of canonical maps?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111014
|
5a81cd1f31013a001a334ef3
|
Group_(mathematics)
|
Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map H → G, i.e. any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map G → G / N.y[›] Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In general, homomorphisms are neither injective nor surjective. Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon.
|
What two forms must homomorphisms take?
|
What two forms must homomorphisms take?
|
[
"What two forms must homomorphisms take?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111015
|
572826002ca10214002d9f12
|
Group_(mathematics)
|
Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups.i[›] For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole.
|
What are usually analyzed by associating groups to them and studying the elements of the corresponding groups?
|
What are usually analyzed by associating groups to them and studying the elements of the corresponding groups?
|
[
"What are usually analyzed by associating groups to them and studying the elements of the corresponding groups?"
] |
{
"text": [
"Mathematical objects"
],
"answer_start": [
58
]
}
|
gem-squad_v2-train-111016
|
572826002ca10214002d9f13
|
Group_(mathematics)
|
Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups.i[›] For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole.
|
Who founded algebraic topology?
|
Who founded algebraic topology?
|
[
"Who founded algebraic topology?"
] |
{
"text": [
"Henri Poincaré"
],
"answer_start": [
198
]
}
|
gem-squad_v2-train-111017
|
572826002ca10214002d9f14
|
Group_(mathematics)
|
Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups.i[›] For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole.
|
What did Henri Poincaré introduce when he established algebraic topology?
|
What did Henri Poincaré introduce when he established algebraic topology?
|
[
"What did Henri Poincaré introduce when he established algebraic topology?"
] |
{
"text": [
"the fundamental group"
],
"answer_start": [
274
]
}
|
gem-squad_v2-train-111018
|
572826002ca10214002d9f15
|
Group_(mathematics)
|
Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups.i[›] For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole.
|
What topological properties translate into properties of groups?
|
What topological properties translate into properties of groups?
|
[
"What topological properties translate into properties of groups?"
] |
{
"text": [
"proximity and continuity"
],
"answer_start": [
357
]
}
|
gem-squad_v2-train-111019
|
572826002ca10214002d9f16
|
Group_(mathematics)
|
Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups.i[›] For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole.
|
What represents elements of the fundamental group?
|
What represents elements of the fundamental group?
|
[
"What represents elements of the fundamental group?"
] |
{
"text": [
"loops"
],
"answer_start": [
489
]
}
|
gem-squad_v2-train-111020
|
5a81ce1331013a001a334ef9
|
Group_(mathematics)
|
Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups.i[›] For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole.
|
Who first applied groups to other mathematical areas?
|
Who first applied groups to other mathematical areas?
|
[
"Who first applied groups to other mathematical areas?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111021
|
5a81ce1331013a001a334efa
|
Group_(mathematics)
|
Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups.i[›] For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole.
|
Which topological properties do not carry over into groups?
|
Which topological properties do not carry over into groups?
|
[
"Which topological properties do not carry over into groups?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111022
|
5a81ce1331013a001a334efb
|
Group_(mathematics)
|
Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups.i[›] For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole.
|
Why is the blue loop considered relevant?
|
Why is the blue loop considered relevant?
|
[
"Why is the blue loop considered relevant?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111023
|
5a81ce1331013a001a334efc
|
Group_(mathematics)
|
Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups.i[›] For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole.
|
What causes the orange loop to shrink to a point?
|
What causes the orange loop to shrink to a point?
|
[
"What causes the orange loop to shrink to a point?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111024
|
5a81ce1331013a001a334efd
|
Group_(mathematics)
|
Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups.i[›] For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole.
|
What results in a finite cycle?
|
What results in a finite cycle?
|
[
"What results in a finite cycle?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111025
|
572826e63acd2414000df5a7
|
Group_(mathematics)
|
In modular arithmetic, two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the remainder of that division. For any modulus, n, the set of integers from 0 to n − 1 forms a group under modular addition: the inverse of any element a is n − a, and 0 is the identity element. This is familiar from the addition of hours on the face of a clock: if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. This is expressed by saying that 9 + 4 equals 1 "modulo 12" or, in symbols,
|
What positive integer is used to divide the sum of two positive integers in modular mathematics?
|
What positive integer is used to divide the sum of two positive integers in modular mathematics?
|
[
"What positive integer is used to divide the sum of two positive integers in modular mathematics?"
] |
{
"text": [
"the modulus"
],
"answer_start": [
103
]
}
|
gem-squad_v2-train-111026
|
572826e63acd2414000df5a8
|
Group_(mathematics)
|
In modular arithmetic, two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the remainder of that division. For any modulus, n, the set of integers from 0 to n − 1 forms a group under modular addition: the inverse of any element a is n − a, and 0 is the identity element. This is familiar from the addition of hours on the face of a clock: if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. This is expressed by saying that 9 + 4 equals 1 "modulo 12" or, in symbols,
|
What results from modular addition?
|
What results from modular addition?
|
[
"What results from modular addition?"
] |
{
"text": [
"the remainder of that division"
],
"answer_start": [
150
]
}
|
gem-squad_v2-train-111027
|
572826e63acd2414000df5a9
|
Group_(mathematics)
|
In modular arithmetic, two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the remainder of that division. For any modulus, n, the set of integers from 0 to n − 1 forms a group under modular addition: the inverse of any element a is n − a, and 0 is the identity element. This is familiar from the addition of hours on the face of a clock: if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. This is expressed by saying that 9 + 4 equals 1 "modulo 12" or, in symbols,
|
What type of device can be use to demonstrate modular addition?
|
What type of device can be use to demonstrate modular addition?
|
[
"What type of device can be use to demonstrate modular addition?"
] |
{
"text": [
"a clock"
],
"answer_start": [
405
]
}
|
gem-squad_v2-train-111028
|
5a81d33331013a001a334f03
|
Group_(mathematics)
|
In modular arithmetic, two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the remainder of that division. For any modulus, n, the set of integers from 0 to n − 1 forms a group under modular addition: the inverse of any element a is n − a, and 0 is the identity element. This is familiar from the addition of hours on the face of a clock: if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. This is expressed by saying that 9 + 4 equals 1 "modulo 12" or, in symbols,
|
How many integers are divided in modular arithmetic?
|
How many integers are divided in modular arithmetic?
|
[
"How many integers are divided in modular arithmetic?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111029
|
5a81d33331013a001a334f04
|
Group_(mathematics)
|
In modular arithmetic, two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the remainder of that division. For any modulus, n, the set of integers from 0 to n − 1 forms a group under modular addition: the inverse of any element a is n − a, and 0 is the identity element. This is familiar from the addition of hours on the face of a clock: if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. This is expressed by saying that 9 + 4 equals 1 "modulo 12" or, in symbols,
|
What are the two integers called?
|
What are the two integers called?
|
[
"What are the two integers called?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111030
|
5a81d33331013a001a334f05
|
Group_(mathematics)
|
In modular arithmetic, two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the remainder of that division. For any modulus, n, the set of integers from 0 to n − 1 forms a group under modular addition: the inverse of any element a is n − a, and 0 is the identity element. This is familiar from the addition of hours on the face of a clock: if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. This is expressed by saying that 9 + 4 equals 1 "modulo 12" or, in symbols,
|
What is the result of the remained of the division?
|
What is the result of the remained of the division?
|
[
"What is the result of the remained of the division?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111031
|
5a81d33331013a001a334f06
|
Group_(mathematics)
|
In modular arithmetic, two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the remainder of that division. For any modulus, n, the set of integers from 0 to n − 1 forms a group under modular addition: the inverse of any element a is n − a, and 0 is the identity element. This is familiar from the addition of hours on the face of a clock: if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. This is expressed by saying that 9 + 4 equals 1 "modulo 12" or, in symbols,
|
What is 9 in modular addition?
|
What is 9 in modular addition?
|
[
"What is 9 in modular addition?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111032
|
5a81d33331013a001a334f07
|
Group_(mathematics)
|
In modular arithmetic, two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the remainder of that division. For any modulus, n, the set of integers from 0 to n − 1 forms a group under modular addition: the inverse of any element a is n − a, and 0 is the identity element. This is familiar from the addition of hours on the face of a clock: if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. This is expressed by saying that 9 + 4 equals 1 "modulo 12" or, in symbols,
|
If the hour hand is on 1 and is advanced by 4 hours, where does it end up?
|
If the hour hand is on 1 and is advanced by 4 hours, where does it end up?
|
[
"If the hour hand is on 1 and is advanced by 4 hours, where does it end up?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111033
|
572827c73acd2414000df5b5
|
Group_(mathematics)
|
For any prime number p, there is also the multiplicative group of integers modulo p. Its elements are the integers 1 to p − 1. The group operation is multiplication modulo p. That is, the usual product is divided by p and the remainder of this division is the result of modular multiplication. For example, if p = 5, there are four group elements 1, 2, 3, 4. In this group, 4 · 4 = 1, because the usual product 16 is equivalent to 1, which divided by 5 yields a remainder of 1. for 5 divides 16 − 1 = 15, denoted
|
What term describes the group of integers related to a prime number?
|
What term describes the group of integers related to a prime number?
|
[
"What term describes the group of integers related to a prime number?"
] |
{
"text": [
"modulo p"
],
"answer_start": [
75
]
}
|
gem-squad_v2-train-111034
|
572827c73acd2414000df5b6
|
Group_(mathematics)
|
For any prime number p, there is also the multiplicative group of integers modulo p. Its elements are the integers 1 to p − 1. The group operation is multiplication modulo p. That is, the usual product is divided by p and the remainder of this division is the result of modular multiplication. For example, if p = 5, there are four group elements 1, 2, 3, 4. In this group, 4 · 4 = 1, because the usual product 16 is equivalent to 1, which divided by 5 yields a remainder of 1. for 5 divides 16 − 1 = 15, denoted
|
What integers are included in modulo p?
|
What integers are included in modulo p?
|
[
"What integers are included in modulo p?"
] |
{
"text": [
"1 to p − 1"
],
"answer_start": [
115
]
}
|
gem-squad_v2-train-111035
|
572827c83acd2414000df5b7
|
Group_(mathematics)
|
For any prime number p, there is also the multiplicative group of integers modulo p. Its elements are the integers 1 to p − 1. The group operation is multiplication modulo p. That is, the usual product is divided by p and the remainder of this division is the result of modular multiplication. For example, if p = 5, there are four group elements 1, 2, 3, 4. In this group, 4 · 4 = 1, because the usual product 16 is equivalent to 1, which divided by 5 yields a remainder of 1. for 5 divides 16 − 1 = 15, denoted
|
How many group elements exist if p=5?
|
How many group elements exist if p=5?
|
[
"How many group elements exist if p=5?"
] |
{
"text": [
"four group elements"
],
"answer_start": [
327
]
}
|
gem-squad_v2-train-111036
|
5a81d44431013a001a334f0d
|
Group_(mathematics)
|
For any prime number p, there is also the multiplicative group of integers modulo p. Its elements are the integers 1 to p − 1. The group operation is multiplication modulo p. That is, the usual product is divided by p and the remainder of this division is the result of modular multiplication. For example, if p = 5, there are four group elements 1, 2, 3, 4. In this group, 4 · 4 = 1, because the usual product 16 is equivalent to 1, which divided by 5 yields a remainder of 1. for 5 divides 16 − 1 = 15, denoted
|
What describes a group of integers related to a modulo?
|
What describes a group of integers related to a modulo?
|
[
"What describes a group of integers related to a modulo?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111037
|
5a81d44431013a001a334f0e
|
Group_(mathematics)
|
For any prime number p, there is also the multiplicative group of integers modulo p. Its elements are the integers 1 to p − 1. The group operation is multiplication modulo p. That is, the usual product is divided by p and the remainder of this division is the result of modular multiplication. For example, if p = 5, there are four group elements 1, 2, 3, 4. In this group, 4 · 4 = 1, because the usual product 16 is equivalent to 1, which divided by 5 yields a remainder of 1. for 5 divides 16 − 1 = 15, denoted
|
What is the result of modular multiplication divided by?
|
What is the result of modular multiplication divided by?
|
[
"What is the result of modular multiplication divided by?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111038
|
5a81d44431013a001a334f0f
|
Group_(mathematics)
|
For any prime number p, there is also the multiplicative group of integers modulo p. Its elements are the integers 1 to p − 1. The group operation is multiplication modulo p. That is, the usual product is divided by p and the remainder of this division is the result of modular multiplication. For example, if p = 5, there are four group elements 1, 2, 3, 4. In this group, 4 · 4 = 1, because the usual product 16 is equivalent to 1, which divided by 5 yields a remainder of 1. for 5 divides 16 − 1 = 15, denoted
|
How many integers does p = 5 have?
|
How many integers does p = 5 have?
|
[
"How many integers does p = 5 have?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111039
|
5a81d44431013a001a334f10
|
Group_(mathematics)
|
For any prime number p, there is also the multiplicative group of integers modulo p. Its elements are the integers 1 to p − 1. The group operation is multiplication modulo p. That is, the usual product is divided by p and the remainder of this division is the result of modular multiplication. For example, if p = 5, there are four group elements 1, 2, 3, 4. In this group, 4 · 4 = 1, because the usual product 16 is equivalent to 1, which divided by 5 yields a remainder of 1. for 5 divides 16 − 1 = 15, denoted
|
What is p divided by?
|
What is p divided by?
|
[
"What is p divided by?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111040
|
5a81d44431013a001a334f11
|
Group_(mathematics)
|
For any prime number p, there is also the multiplicative group of integers modulo p. Its elements are the integers 1 to p − 1. The group operation is multiplication modulo p. That is, the usual product is divided by p and the remainder of this division is the result of modular multiplication. For example, if p = 5, there are four group elements 1, 2, 3, 4. In this group, 4 · 4 = 1, because the usual product 16 is equivalent to 1, which divided by 5 yields a remainder of 1. for 5 divides 16 − 1 = 15, denoted
|
What is 16 multiplied by 5 If p=5?
|
What is 16 multiplied by 5 If p=5?
|
[
"What is 16 multiplied by 5 If p=5?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111041
|
572829024b864d1900164612
|
Group_(mathematics)
|
In the groups Z/nZ introduced above, the element 1 is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are 1. Any cyclic group with n elements is isomorphic to this group. A second example for cyclic groups is the group of n-th complex roots of unity, given by complex numbers z satisfying zn = 1. These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n = 6. The group operation is multiplication of complex numbers. In the picture, multiplying with z corresponds to a counter-clockwise rotation by 60°. Using some field theory, the group Fp× can be shown to be cyclic: for example, if p = 5, 3 is a generator since 31 = 3, 32 = 9 ≡ 4, 33 ≡ 2, and 34 ≡ 1.
|
What is a group labeled when the element 1 is primitive?
|
What is a group labeled when the element 1 is primitive?
|
[
"What is a group labeled when the element 1 is primitive?"
] |
{
"text": [
"cyclic"
],
"answer_start": [
85
]
}
|
gem-squad_v2-train-111042
|
572829024b864d1900164613
|
Group_(mathematics)
|
In the groups Z/nZ introduced above, the element 1 is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are 1. Any cyclic group with n elements is isomorphic to this group. A second example for cyclic groups is the group of n-th complex roots of unity, given by complex numbers z satisfying zn = 1. These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n = 6. The group operation is multiplication of complex numbers. In the picture, multiplying with z corresponds to a counter-clockwise rotation by 60°. Using some field theory, the group Fp× can be shown to be cyclic: for example, if p = 5, 3 is a generator since 31 = 3, 32 = 9 ≡ 4, 33 ≡ 2, and 34 ≡ 1.
|
What group is isomorphic to cyclic groups?
|
What group is isomorphic to cyclic groups?
|
[
"What group is isomorphic to cyclic groups?"
] |
{
"text": [
"Any cyclic group with n elements"
],
"answer_start": [
164
]
}
|
gem-squad_v2-train-111043
|
572829024b864d1900164614
|
Group_(mathematics)
|
In the groups Z/nZ introduced above, the element 1 is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are 1. Any cyclic group with n elements is isomorphic to this group. A second example for cyclic groups is the group of n-th complex roots of unity, given by complex numbers z satisfying zn = 1. These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n = 6. The group operation is multiplication of complex numbers. In the picture, multiplying with z corresponds to a counter-clockwise rotation by 60°. Using some field theory, the group Fp× can be shown to be cyclic: for example, if p = 5, 3 is a generator since 31 = 3, 32 = 9 ≡ 4, 33 ≡ 2, and 34 ≡ 1.
|
What example of cyclic group satisfies the express of zn = 1?
|
What example of cyclic group satisfies the express of zn = 1?
|
[
"What example of cyclic group satisfies the express of zn = 1?"
] |
{
"text": [
"the group of n-th complex roots of unity"
],
"answer_start": [
264
]
}
|
gem-squad_v2-train-111044
|
5a81e1e031013a001a334f61
|
Group_(mathematics)
|
In the groups Z/nZ introduced above, the element 1 is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are 1. Any cyclic group with n elements is isomorphic to this group. A second example for cyclic groups is the group of n-th complex roots of unity, given by complex numbers z satisfying zn = 1. These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n = 6. The group operation is multiplication of complex numbers. In the picture, multiplying with z corresponds to a counter-clockwise rotation by 60°. Using some field theory, the group Fp× can be shown to be cyclic: for example, if p = 5, 3 is a generator since 31 = 3, 32 = 9 ≡ 4, 33 ≡ 2, and 34 ≡ 1.
|
A cyclic group without n elements is considered what?
|
A cyclic group without n elements is considered what?
|
[
"A cyclic group without n elements is considered what?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111045
|
5a81e1e031013a001a334f62
|
Group_(mathematics)
|
In the groups Z/nZ introduced above, the element 1 is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are 1. Any cyclic group with n elements is isomorphic to this group. A second example for cyclic groups is the group of n-th complex roots of unity, given by complex numbers z satisfying zn = 1. These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n = 6. The group operation is multiplication of complex numbers. In the picture, multiplying with z corresponds to a counter-clockwise rotation by 60°. Using some field theory, the group Fp× can be shown to be cyclic: for example, if p = 5, 3 is a generator since 31 = 3, 32 = 9 ≡ 4, 33 ≡ 2, and 34 ≡ 1.
|
What element is not primitive in groups Z/nZ?
|
What element is not primitive in groups Z/nZ?
|
[
"What element is not primitive in groups Z/nZ?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111046
|
5a81e1e031013a001a334f63
|
Group_(mathematics)
|
In the groups Z/nZ introduced above, the element 1 is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are 1. Any cyclic group with n elements is isomorphic to this group. A second example for cyclic groups is the group of n-th complex roots of unity, given by complex numbers z satisfying zn = 1. These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n = 6. The group operation is multiplication of complex numbers. In the picture, multiplying with z corresponds to a counter-clockwise rotation by 60°. Using some field theory, the group Fp× can be shown to be cyclic: for example, if p = 5, 3 is a generator since 31 = 3, 32 = 9 ≡ 4, 33 ≡ 2, and 34 ≡ 1.
|
What are groups called when element 1 is not primitive?
|
What are groups called when element 1 is not primitive?
|
[
"What are groups called when element 1 is not primitive?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111047
|
5a81e1e031013a001a334f64
|
Group_(mathematics)
|
In the groups Z/nZ introduced above, the element 1 is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are 1. Any cyclic group with n elements is isomorphic to this group. A second example for cyclic groups is the group of n-th complex roots of unity, given by complex numbers z satisfying zn = 1. These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n = 6. The group operation is multiplication of complex numbers. In the picture, multiplying with z corresponds to a counter-clockwise rotation by 60°. Using some field theory, the group Fp× can be shown to be cyclic: for example, if p = 5, 3 is a generator since 31 = 3, 32 = 9 ≡ 4, 33 ≡ 2, and 34 ≡ 1.
|
What rotation results in multiplying but n?
|
What rotation results in multiplying but n?
|
[
"What rotation results in multiplying but n?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111048
|
5a81e1e031013a001a334f65
|
Group_(mathematics)
|
In the groups Z/nZ introduced above, the element 1 is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are 1. Any cyclic group with n elements is isomorphic to this group. A second example for cyclic groups is the group of n-th complex roots of unity, given by complex numbers z satisfying zn = 1. These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n = 6. The group operation is multiplication of complex numbers. In the picture, multiplying with z corresponds to a counter-clockwise rotation by 60°. Using some field theory, the group Fp× can be shown to be cyclic: for example, if p = 5, 3 is a generator since 31 = 3, 32 = 9 ≡ 4, 33 ≡ 2, and 34 ≡ 1.
|
What are numbers on an irregular n-gon?
|
What are numbers on an irregular n-gon?
|
[
"What are numbers on an irregular n-gon?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111049
|
57282a4b2ca10214002d9fc0
|
Group_(mathematics)
|
Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry.t[›] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.
|
What are groups consisting of symmetries of given arithmetic concepts?
|
What are groups consisting of symmetries of given arithmetic concepts?
|
[
"What are groups consisting of symmetries of given arithmetic concepts?"
] |
{
"text": [
"Symmetry groups"
],
"answer_start": [
0
]
}
|
gem-squad_v2-train-111050
|
57282a4b2ca10214002d9fc1
|
Group_(mathematics)
|
Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry.t[›] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.
|
What term describes the introductory symmetry group of the square?
|
What term describes the introductory symmetry group of the square?
|
[
"What term describes the introductory symmetry group of the square?"
] |
{
"text": [
"geometric nature"
],
"answer_start": [
93
]
}
|
gem-squad_v2-train-111051
|
57282a4b2ca10214002d9fc2
|
Group_(mathematics)
|
Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry.t[›] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.
|
What can regarded as the study of symmetry?
|
What can regarded as the study of symmetry?
|
[
"What can regarded as the study of symmetry?"
] |
{
"text": [
"group theory"
],
"answer_start": [
254
]
}
|
gem-squad_v2-train-111052
|
57282a4b2ca10214002d9fc3
|
Group_(mathematics)
|
Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry.t[›] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.
|
What is the name of the rule that must be met for a group operation to occur?
|
What is the name of the rule that must be met for a group operation to occur?
|
[
"What is the name of the rule that must be met for a group operation to occur?"
] |
{
"text": [
"group law"
],
"answer_start": [
532
]
}
|
gem-squad_v2-train-111053
|
57282a4b2ca10214002d9fc4
|
Group_(mathematics)
|
Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry.t[›] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.
|
The group pattern is connected to the structure of the target by what behavior?
|
The group pattern is connected to the structure of the target by what behavior?
|
[
"The group pattern is connected to the structure of the target by what behavior?"
] |
{
"text": [
"group action"
],
"answer_start": [
724
]
}
|
gem-squad_v2-train-111054
|
5a81e2e631013a001a334f6b
|
Group_(mathematics)
|
Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry.t[›] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.
|
What do symmetry groups make?
|
What do symmetry groups make?
|
[
"What do symmetry groups make?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111055
|
5a81e2e631013a001a334f6c
|
Group_(mathematics)
|
Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry.t[›] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.
|
Which equation is are used for the square?
|
Which equation is are used for the square?
|
[
"Which equation is are used for the square?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111056
|
5a81e2e631013a001a334f6d
|
Group_(mathematics)
|
Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry.t[›] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.
|
What is the study of geometrical or analytical objects called?
|
What is the study of geometrical or analytical objects called?
|
[
"What is the study of geometrical or analytical objects called?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111057
|
5a81e2e631013a001a334f6e
|
Group_(mathematics)
|
Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry.t[›] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.
|
What makes the study of symmetries easier?
|
What makes the study of symmetries easier?
|
[
"What makes the study of symmetries easier?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111058
|
5a81e2e631013a001a334f6f
|
Group_(mathematics)
|
Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry.t[›] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.
|
What does group action do to the tiling?
|
What does group action do to the tiling?
|
[
"What does group action do to the tiling?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111059
|
57282d592ca10214002d9ff0
|
Group_(mathematics)
|
Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.
|
What aids in predicting changes of physical traits?
|
What aids in predicting changes of physical traits?
|
[
"What aids in predicting changes of physical traits?"
] |
{
"text": [
"group theory"
],
"answer_start": [
10
]
}
|
gem-squad_v2-train-111060
|
57282d592ca10214002d9ff1
|
Group_(mathematics)
|
Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.
|
What stage of a physical transformation can group theory be used to make prediction?
|
What stage of a physical transformation can group theory be used to make prediction?
|
[
"What stage of a physical transformation can group theory be used to make prediction?"
] |
{
"text": [
"phase transition"
],
"answer_start": [
111
]
}
|
gem-squad_v2-train-111061
|
57282d592ca10214002d9ff2
|
Group_(mathematics)
|
Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.
|
What temperature causes the change of ferroelectric materials?
|
What temperature causes the change of ferroelectric materials?
|
[
"What temperature causes the change of ferroelectric materials?"
] |
{
"text": [
"Curie temperature"
],
"answer_start": [
305
]
}
|
gem-squad_v2-train-111062
|
57282d592ca10214002d9ff3
|
Group_(mathematics)
|
Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.
|
What term describes the vibrational lattice mode that turns to 0 frequency at the change?
|
What term describes the vibrational lattice mode that turns to 0 frequency at the change?
|
[
"What term describes the vibrational lattice mode that turns to 0 frequency at the change?"
] |
{
"text": [
"soft phonon mode"
],
"answer_start": [
461
]
}
|
gem-squad_v2-train-111063
|
5a81e3c131013a001a334f75
|
Group_(mathematics)
|
Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.
|
What predicts the changes in group theory?
|
What predicts the changes in group theory?
|
[
"What predicts the changes in group theory?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111064
|
5a81e3c131013a001a334f76
|
Group_(mathematics)
|
Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.
|
Which stage can physical properties be used to make predictions about group theory?
|
Which stage can physical properties be used to make predictions about group theory?
|
[
"Which stage can physical properties be used to make predictions about group theory?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111065
|
5a81e3c131013a001a334f77
|
Group_(mathematics)
|
Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.
|
What temperature causes a change from ferroelectric to paraelectric?
|
What temperature causes a change from ferroelectric to paraelectric?
|
[
"What temperature causes a change from ferroelectric to paraelectric?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111066
|
5a81e3c131013a001a334f78
|
Group_(mathematics)
|
Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.
|
What is the frequency of a paraelectric state before transition?
|
What is the frequency of a paraelectric state before transition?
|
[
"What is the frequency of a paraelectric state before transition?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111067
|
5a81e3c131013a001a334f79
|
Group_(mathematics)
|
Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.
|
What is the name of the mode that moves material from a low-symmetry state to a high one?
|
What is the name of the mode that moves material from a low-symmetry state to a high one?
|
[
"What is the name of the mode that moves material from a low-symmetry state to a high one?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111068
|
5728362a3acd2414000df70f
|
Group_(mathematics)
|
Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved.u[›] Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.
|
What are examples of finite symmetry groups used in coding theory?
|
What are examples of finite symmetry groups used in coding theory?
|
[
"What are examples of finite symmetry groups used in coding theory?"
] |
{
"text": [
"Mathieu groups"
],
"answer_start": [
35
]
}
|
gem-squad_v2-train-111069
|
5728362a3acd2414000df710
|
Group_(mathematics)
|
Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved.u[›] Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.
|
What is used for error correction of transferred data?
|
What is used for error correction of transferred data?
|
[
"What is used for error correction of transferred data?"
] |
{
"text": [
"coding theory"
],
"answer_start": [
62
]
}
|
gem-squad_v2-train-111070
|
5728362a3acd2414000df711
|
Group_(mathematics)
|
Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved.u[›] Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.
|
What describes functions having antiderivatives of a prescribed form?
|
What describes functions having antiderivatives of a prescribed form?
|
[
"What describes functions having antiderivatives of a prescribed form?"
] |
{
"text": [
"differential Galois theory"
],
"answer_start": [
185
]
}
|
gem-squad_v2-train-111071
|
5728362a3acd2414000df712
|
Group_(mathematics)
|
Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved.u[›] Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.
|
What concept investigates geometric elements that stay stable under group action?
|
What concept investigates geometric elements that stay stable under group action?
|
[
"What concept investigates geometric elements that stay stable under group action?"
] |
{
"text": [
"invariant theory"
],
"answer_start": [
487
]
}
|
gem-squad_v2-train-111072
|
5a81e49731013a001a334f89
|
Group_(mathematics)
|
Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved.u[›] Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.
|
What theory is error correction applied to?
|
What theory is error correction applied to?
|
[
"What theory is error correction applied to?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111073
|
5a81e49731013a001a334f8a
|
Group_(mathematics)
|
Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved.u[›] Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.
|
Which form of finite symmetry groups are useless in coding theory?
|
Which form of finite symmetry groups are useless in coding theory?
|
[
"Which form of finite symmetry groups are useless in coding theory?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111074
|
5a81e49731013a001a334f8b
|
Group_(mathematics)
|
Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved.u[›] Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.
|
Mathieu groups have what kind of prescribed form?
|
Mathieu groups have what kind of prescribed form?
|
[
"Mathieu groups have what kind of prescribed form?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111075
|
5a81e49731013a001a334f8c
|
Group_(mathematics)
|
Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved.u[›] Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.
|
What stable properties does the Galois theory study?
|
What stable properties does the Galois theory study?
|
[
"What stable properties does the Galois theory study?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111076
|
5a81e49731013a001a334f8d
|
Group_(mathematics)
|
Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved.u[›] Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.
|
Geometric Invariant theory studies what kind of equations?
|
Geometric Invariant theory studies what kind of equations?
|
[
"Geometric Invariant theory studies what kind of equations?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111077
|
5728372a4b864d190016475c
|
Group_(mathematics)
|
Matrix groups consist of matrices together with matrix multiplication. The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries. Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics.
|
What groups combine matrices with matrix multiplication?
|
What groups combine matrices with matrix multiplication?
|
[
"What groups combine matrices with matrix multiplication?"
] |
{
"text": [
"Matrix groups"
],
"answer_start": [
0
]
}
|
gem-squad_v2-train-111078
|
5728372a4b864d190016475d
|
Group_(mathematics)
|
Matrix groups consist of matrices together with matrix multiplication. The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries. Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics.
|
What includes all invertible n-by-n matrices with real entries?
|
What includes all invertible n-by-n matrices with real entries?
|
[
"What includes all invertible n-by-n matrices with real entries?"
] |
{
"text": [
"The general linear group"
],
"answer_start": [
71
]
}
|
gem-squad_v2-train-111079
|
5728372a4b864d190016475e
|
Group_(mathematics)
|
Matrix groups consist of matrices together with matrix multiplication. The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries. Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics.
|
What term describes subgroups of the general linear group?
|
What term describes subgroups of the general linear group?
|
[
"What term describes subgroups of the general linear group?"
] |
{
"text": [
"matrix groups"
],
"answer_start": [
200
]
}
|
gem-squad_v2-train-111080
|
5728372a4b864d190016475f
|
Group_(mathematics)
|
Matrix groups consist of matrices together with matrix multiplication. The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries. Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics.
|
What matrix group portrays all possible rotations in n dimensions?
|
What matrix group portrays all possible rotations in n dimensions?
|
[
"What matrix group portrays all possible rotations in n dimensions?"
] |
{
"text": [
"the special orthogonal group SO(n)"
],
"answer_start": [
355
]
}
|
gem-squad_v2-train-111081
|
5728372a4b864d1900164760
|
Group_(mathematics)
|
Matrix groups consist of matrices together with matrix multiplication. The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries. Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics.
|
Rotation matrix groups are utilized in computer graphics with what concept?
|
Rotation matrix groups are utilized in computer graphics with what concept?
|
[
"Rotation matrix groups are utilized in computer graphics with what concept?"
] |
{
"text": [
"Euler angles"
],
"answer_start": [
448
]
}
|
gem-squad_v2-train-111082
|
5a81e5d531013a001a334f93
|
Group_(mathematics)
|
Matrix groups consist of matrices together with matrix multiplication. The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries. Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics.
|
What are matrices and matrix groups combined called?
|
What are matrices and matrix groups combined called?
|
[
"What are matrices and matrix groups combined called?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111083
|
5a81e5d531013a001a334f94
|
Group_(mathematics)
|
Matrix groups consist of matrices together with matrix multiplication. The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries. Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics.
|
The general linear group describes all possible what?
|
The general linear group describes all possible what?
|
[
"The general linear group describes all possible what?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111084
|
5a81e5d531013a001a334f95
|
Group_(mathematics)
|
Matrix groups consist of matrices together with matrix multiplication. The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries. Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics.
|
What is an example of a large matrix group?
|
What is an example of a large matrix group?
|
[
"What is an example of a large matrix group?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111085
|
5a81e5d531013a001a334f96
|
Group_(mathematics)
|
Matrix groups consist of matrices together with matrix multiplication. The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries. Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics.
|
What are subgroups of the special orthogonal group called?
|
What are subgroups of the special orthogonal group called?
|
[
"What are subgroups of the special orthogonal group called?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111086
|
5a81e5d531013a001a334f97
|
Group_(mathematics)
|
Matrix groups consist of matrices together with matrix multiplication. The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries. Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics.
|
What type of angles are made of all invertible n-by-b matrices?
|
What type of angles are made of all invertible n-by-b matrices?
|
[
"What type of angles are made of all invertible n-by-b matrices?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111087
|
572838983acd2414000df749
|
Group_(mathematics)
|
Exchanging "+" and "−" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e. solutions expressible using solely addition, multiplication, and roots similar to the formula above.
|
What degree does not include simple formulas for cubic and quatric equations?
|
What degree does not include simple formulas for cubic and quatric equations?
|
[
"What degree does not include simple formulas for cubic and quatric equations?"
] |
{
"text": [
"degree 5 and higher"
],
"answer_start": [
233
]
}
|
gem-squad_v2-train-111088
|
572838983acd2414000df74a
|
Group_(mathematics)
|
Exchanging "+" and "−" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e. solutions expressible using solely addition, multiplication, and roots similar to the formula above.
|
What concept is associated with the solvability of polynomials?
|
What concept is associated with the solvability of polynomials?
|
[
"What concept is associated with the solvability of polynomials?"
] |
{
"text": [
"Abstract properties of Galois groups"
],
"answer_start": [
254
]
}
|
gem-squad_v2-train-111089
|
572838983acd2414000df74b
|
Group_(mathematics)
|
Exchanging "+" and "−" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e. solutions expressible using solely addition, multiplication, and roots similar to the formula above.
|
What are used to express the solutions of polynomials?
|
What are used to express the solutions of polynomials?
|
[
"What are used to express the solutions of polynomials?"
] |
{
"text": [
"radicals"
],
"answer_start": [
431
]
}
|
gem-squad_v2-train-111090
|
5a81f51431013a001a334fd9
|
Group_(mathematics)
|
Exchanging "+" and "−" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e. solutions expressible using solely addition, multiplication, and roots similar to the formula above.
|
What degree must it be lower than to have simple formulas?
|
What degree must it be lower than to have simple formulas?
|
[
"What degree must it be lower than to have simple formulas?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111091
|
5a81f51431013a001a334fda
|
Group_(mathematics)
|
Exchanging "+" and "−" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e. solutions expressible using solely addition, multiplication, and roots similar to the formula above.
|
What give criterion for abstract properties?
|
What give criterion for abstract properties?
|
[
"What give criterion for abstract properties?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111092
|
5a81f51431013a001a334fdb
|
Group_(mathematics)
|
Exchanging "+" and "−" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e. solutions expressible using solely addition, multiplication, and roots similar to the formula above.
|
Which group cannot be expressed using radicals?
|
Which group cannot be expressed using radicals?
|
[
"Which group cannot be expressed using radicals?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111093
|
5a81f51431013a001a334fdc
|
Group_(mathematics)
|
Exchanging "+" and "−" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e. solutions expressible using solely addition, multiplication, and roots similar to the formula above.
|
What is an example of a complex operation?
|
What is an example of a complex operation?
|
[
"What is an example of a complex operation?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111094
|
5a81f51431013a001a334fdd
|
Group_(mathematics)
|
Exchanging "+" and "−" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e. solutions expressible using solely addition, multiplication, and roots similar to the formula above.
|
What do cubic and quartic equations not have?
|
What do cubic and quartic equations not have?
|
[
"What do cubic and quartic equations not have?"
] |
{
"text": [],
"answer_start": []
}
|
gem-squad_v2-train-111095
|
5728396e2ca10214002da110
|
Group_(mathematics)
|
A group is called finite if it has a finite number of elements. The number of elements is called the order of the group. An important class is the symmetric groups SN, the groups of permutations of N letters. For example, the symmetric group on 3 letters S3 is the group consisting of all possible orderings of the three letters ABC, i.e. contains the elements ABC, ACB, ..., up to CBA, in total 6 (or 3 factorial) elements. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group SN for a suitable integer N (Cayley's theorem). Parallel to the group of symmetries of the square above, S3 can also be interpreted as the group of symmetries of an equilateral triangle.
|
What does a finite group include?
|
What does a finite group include?
|
[
"What does a finite group include?"
] |
{
"text": [
"a finite number of elements"
],
"answer_start": [
35
]
}
|
gem-squad_v2-train-111096
|
5728396e2ca10214002da111
|
Group_(mathematics)
|
A group is called finite if it has a finite number of elements. The number of elements is called the order of the group. An important class is the symmetric groups SN, the groups of permutations of N letters. For example, the symmetric group on 3 letters S3 is the group consisting of all possible orderings of the three letters ABC, i.e. contains the elements ABC, ACB, ..., up to CBA, in total 6 (or 3 factorial) elements. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group SN for a suitable integer N (Cayley's theorem). Parallel to the group of symmetries of the square above, S3 can also be interpreted as the group of symmetries of an equilateral triangle.
|
What is the number of elements in a group named?
|
What is the number of elements in a group named?
|
[
"What is the number of elements in a group named?"
] |
{
"text": [
"the order of the group"
],
"answer_start": [
97
]
}
|
gem-squad_v2-train-111097
|
5728396e2ca10214002da112
|
Group_(mathematics)
|
A group is called finite if it has a finite number of elements. The number of elements is called the order of the group. An important class is the symmetric groups SN, the groups of permutations of N letters. For example, the symmetric group on 3 letters S3 is the group consisting of all possible orderings of the three letters ABC, i.e. contains the elements ABC, ACB, ..., up to CBA, in total 6 (or 3 factorial) elements. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group SN for a suitable integer N (Cayley's theorem). Parallel to the group of symmetries of the square above, S3 can also be interpreted as the group of symmetries of an equilateral triangle.
|
What type of class has a finite group that can be expressed as a subgroup of a symmetric group?
|
What type of class has a finite group that can be expressed as a subgroup of a symmetric group?
|
[
"What type of class has a finite group that can be expressed as a subgroup of a symmetric group?"
] |
{
"text": [
"fundamental"
],
"answer_start": [
439
]
}
|
gem-squad_v2-train-111098
|
5728396e2ca10214002da113
|
Group_(mathematics)
|
A group is called finite if it has a finite number of elements. The number of elements is called the order of the group. An important class is the symmetric groups SN, the groups of permutations of N letters. For example, the symmetric group on 3 letters S3 is the group consisting of all possible orderings of the three letters ABC, i.e. contains the elements ABC, ACB, ..., up to CBA, in total 6 (or 3 factorial) elements. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group SN for a suitable integer N (Cayley's theorem). Parallel to the group of symmetries of the square above, S3 can also be interpreted as the group of symmetries of an equilateral triangle.
|
What can be described as the group of symmetries of an equilateral triangle?
|
What can be described as the group of symmetries of an equilateral triangle?
|
[
"What can be described as the group of symmetries of an equilateral triangle?"
] |
{
"text": [
"S3"
],
"answer_start": [
636
]
}
|
gem-squad_v2-train-111099
|
5a81f74231013a001a334fed
|
Group_(mathematics)
|
A group is called finite if it has a finite number of elements. The number of elements is called the order of the group. An important class is the symmetric groups SN, the groups of permutations of N letters. For example, the symmetric group on 3 letters S3 is the group consisting of all possible orderings of the three letters ABC, i.e. contains the elements ABC, ACB, ..., up to CBA, in total 6 (or 3 factorial) elements. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group SN for a suitable integer N (Cayley's theorem). Parallel to the group of symmetries of the square above, S3 can also be interpreted as the group of symmetries of an equilateral triangle.
|
What is called the order of the group?
|
What is called the order of the group?
|
[
"What is called the order of the group?"
] |
{
"text": [],
"answer_start": []
}
|
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