gem_id stringlengths 20 25 | id stringlengths 24 24 | title stringlengths 3 59 | context stringlengths 151 3.71k | question stringlengths 1 270 | target stringlengths 1 270 | 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 r... | 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 r... | 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 r... | 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 r... | 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 r... | 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 r... | 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 / ... | 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 / ... | 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 / ... | 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 / ... | 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 / ... | 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 / ... | 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 / ... | 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 / ... | 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 / ... | 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 connec... | 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 connec... | 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 connec... | 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 connec... | 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 connec... | 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 connec... | 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 connec... | 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 connec... | 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 connec... | 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 connec... | 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... | 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... | 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... | 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... | 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... | 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... | 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... | 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... | 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, the... | 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, the... | 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, the... | 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, the... | 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, the... | 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, the... | 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, the... | 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, the... | 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 compl... | 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 compl... | 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 compl... | 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 compl... | 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 compl... | 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 compl... | 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 compl... | 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 compl... | 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[›] Symme... | 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[›] Symme... | 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[›] Symme... | 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[›] Symme... | 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[›] Symme... | 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[›] Symme... | 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[›] Symme... | 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[›] Symme... | 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[›] Symme... | 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[›] Symme... | 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 temperatu... | 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 temperatu... | 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 temperatu... | 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 temperatu... | 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 temperatu... | 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 temperatu... | 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 temperatu... | 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 temperatu... | 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 temperatu... | 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 ... | 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 ... | 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 ... | 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 ... | 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 ... | 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 ... | 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 ... | 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 ... | 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 ... | 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.... | 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.... | 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.... | 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.... | 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.... | 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.... | 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.... | 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.... | 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.... | 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.... | 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 (... | 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 (... | 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 (... | 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 (... | 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 (... | 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 (... | 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 (... | 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 (... | 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... | 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... | 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... | 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... | 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... | 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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