id stringlengths 24 24 | title stringclasses 442
values | context stringlengths 151 3.71k | question stringlengths 12 270 | answers dict |
|---|---|---|---|---|
5728106fff5b5019007d9c3c | Group_(mathematics) | The convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques (1870). Walther von Dyck (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definit... | What theory did Hermann Weyl study in addition to locally compact groups? | {
"answer_start": [
636
],
"text": [
"The theory of Lie groups"
]
} |
5728106fff5b5019007d9c3d | Group_(mathematics) | The convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques (1870). Walther von Dyck (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definit... | Who initially molded the theory of algebraic groups? | {
"answer_start": [
841
],
"text": [
"Claude Chevalley"
]
} |
572811cc2ca10214002d9d22 | Group_(mathematics) | The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, classified all finite simple groups in 1982. This project exceeded previous ma... | Where did the group of theorists first meet? | {
"answer_start": [
0
],
"text": [
"The University of Chicago"
]
} |
572811cc2ca10214002d9d23 | Group_(mathematics) | The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, classified all finite simple groups in 1982. This project exceeded previous ma... | What university class year belonged to the group of theorists? | {
"answer_start": [
28
],
"text": [
"1960–61"
]
} |
572811cc2ca10214002d9d24 | Group_(mathematics) | The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, classified all finite simple groups in 1982. This project exceeded previous ma... | What did the group of theorists classify in 1982? | {
"answer_start": [
253
],
"text": [
"all finite simple groups"
]
} |
572811cc2ca10214002d9d25 | Group_(mathematics) | The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, classified all finite simple groups in 1982. This project exceeded previous ma... | What two factors describe the large size of the project? | {
"answer_start": [
369
],
"text": [
"length of proof and number of researchers."
]
} |
572812cd2ca10214002d9d48 | Group_(mathematics) | To understand groups beyond the level of mere symbolic manipulations as above, more structural concepts have to be employed.c[›] There is a conceptual principle underlying all of the following notions: to take advantage of the structure offered by groups (which sets, being "structureless", do not have), constructions r... | What ideas are used to understand groups beyond symbols? | {
"answer_start": [
84
],
"text": [
"structural concepts"
]
} |
572812cd2ca10214002d9d49 | Group_(mathematics) | To understand groups beyond the level of mere symbolic manipulations as above, more structural concepts have to be employed.c[›] There is a conceptual principle underlying all of the following notions: to take advantage of the structure offered by groups (which sets, being "structureless", do not have), constructions r... | What has to be compatible with the group operation? | {
"answer_start": [
305
],
"text": [
"constructions related to groups"
]
} |
572812cd2ca10214002d9d4a | Group_(mathematics) | To understand groups beyond the level of mere symbolic manipulations as above, more structural concepts have to be employed.c[›] There is a conceptual principle underlying all of the following notions: to take advantage of the structure offered by groups (which sets, being "structureless", do not have), constructions r... | What concept describes groups that can be related to each other via functions? | {
"answer_start": [
533
],
"text": [
"group homomorphisms"
]
} |
57281407ff5b5019007d9ca6 | Group_(mathematics) | Two groups G and H are called isomorphic if there exist group homomorphisms a: G → H and b: H → G, such that applying the two functions one after another in each of the two possible orders gives the identity functions of G and H. That is, a(b(h)) = h and b(a(g)) = g for any g in G and h in H. From an abstract point of ... | What are two groups called if they include homomorphisms? | {
"answer_start": [
30
],
"text": [
"isomorphic"
]
} |
57281407ff5b5019007d9ca7 | Group_(mathematics) | Two groups G and H are called isomorphic if there exist group homomorphisms a: G → H and b: H → G, such that applying the two functions one after another in each of the two possible orders gives the identity functions of G and H. That is, a(b(h)) = h and b(a(g)) = g for any g in G and h in H. From an abstract point of ... | What similar element do isomorphic groups carry? | {
"answer_start": [
326
],
"text": [
"isomorphic groups"
]
} |
57281407ff5b5019007d9ca8 | Group_(mathematics) | Two groups G and H are called isomorphic if there exist group homomorphisms a: G → H and b: H → G, such that applying the two functions one after another in each of the two possible orders gives the identity functions of G and H. That is, a(b(h)) = h and b(a(g)) = g for any g in G and h in H. From an abstract point of ... | How can showing the second equality yields the first prove the concept of isomorphic groups? | {
"answer_start": [
489
],
"text": [
"applying a to the first equality yields the second"
]
} |
572820982ca10214002d9e86 | Group_(mathematics) | In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (n... | What is composed of two rotations? | {
"answer_start": [
179
],
"text": [
"a rotation"
]
} |
572820982ca10214002d9e87 | Group_(mathematics) | In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (n... | What rotation can a rotation be reversed by? | {
"answer_start": [
232
],
"text": [
"inverse"
]
} |
572820982ca10214002d9e88 | Group_(mathematics) | In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (n... | What essential condition must be met for a subset of a group to be a subgroup? | {
"answer_start": [
381
],
"text": [
"The subgroup test"
]
} |
572821b7ff5b5019007d9dac | Group_(mathematics) | In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in D4 above, once a reflection is performed, the square never gets back to the r2 configuration by just applying the rotation operations (and no further reflections), i.e. the rotati... | When is it preferable to consider two group elements to be the same? | {
"answer_start": [
338
],
"text": [
"irrelevant to the question whether a reflection has been performed"
]
} |
572821b7ff5b5019007d9dad | Group_(mathematics) | In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in D4 above, once a reflection is performed, the square never gets back to the r2 configuration by just applying the rotation operations (and no further reflections), i.e. the rotati... | What should not be considered when asking if a reflection has been performed? | {
"answer_start": [
314
],
"text": [
"rotation operations"
]
} |
572821b7ff5b5019007d9dae | Group_(mathematics) | In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in D4 above, once a reflection is performed, the square never gets back to the r2 configuration by just applying the rotation operations (and no further reflections), i.e. the rotati... | What number sets are used to show how subgroups can be seen as translations of the larger group? | {
"answer_start": [
406
],
"text": [
"Cosets"
]
} |
572823544b864d1900164546 | Group_(mathematics) | This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group G: (gN) • (hN) = (gh)N for all g and h in G. This definition is motivated by the idea (itself an instance of general structural considerations outlined above) that the map G → G / N that associates to ... | What is another term for coset multiplication? | {
"answer_start": [
79
],
"text": [
"coset addition"
]
} |
572823544b864d1900164547 | Group_(mathematics) | This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group G: (gN) • (hN) = (gh)N for all g and h in G. This definition is motivated by the idea (itself an instance of general structural considerations outlined above) that the map G → G / N that associates to ... | What group operation is taken by the set from the original group? | {
"answer_start": [
54
],
"text": [
"coset multiplication"
]
} |
572823544b864d1900164548 | Group_(mathematics) | This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group G: (gN) • (hN) = (gh)N for all g and h in G. This definition is motivated by the idea (itself an instance of general structural considerations outlined above) that the map G → G / N that associates to ... | What group does the inverse of gN include? | {
"answer_start": [
523
],
"text": [
"quotient group"
]
} |
572823f82ca10214002d9ec6 | 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 groups can be combined to describe every group? | {
"answer_start": [
107
],
"text": [
"is the quotient of the free group over the generators of the group"
]
} |
572823f82ca10214002d9ec7 | 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 group can be the quotient of the free group over the generators of the group? | {
"answer_start": [
97
],
"text": [
"any group"
]
} |
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? | {
"answer_start": [
403
],
"text": [
"every symmetry of the square"
]
} |
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? | {
"answer_start": [
93
],
"text": [
"injective map"
]
} |
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? | {
"answer_start": [
224
],
"text": [
"surjective maps"
]
} |
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? | {
"answer_start": [
298
],
"text": [
"canonical map"
]
} |
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? | {
"answer_start": [
599
],
"text": [
"the first isomorphism theorem"
]
} |
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? | {
"answer_start": [
58
],
"text": [
"Mathematical objects"
]
} |
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? | {
"answer_start": [
198
],
"text": [
"Henri Poincaré"
]
} |
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? | {
"answer_start": [
274
],
"text": [
"the fundamental group"
]
} |
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? | {
"answer_start": [
357
],
"text": [
"proximity and continuity"
]
} |
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? | {
"answer_start": [
489
],
"text": [
"loops"
]
} |
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? | {
"answer_start": [
103
],
"text": [
"the modulus"
]
} |
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? | {
"answer_start": [
150
],
"text": [
"the remainder of that division"
]
} |
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? | {
"answer_start": [
405
],
"text": [
"a clock"
]
} |
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? | {
"answer_start": [
75
],
"text": [
"modulo p"
]
} |
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? | {
"answer_start": [
115
],
"text": [
"1 to p − 1"
]
} |
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? | {
"answer_start": [
327
],
"text": [
"four group elements"
]
} |
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? | {
"answer_start": [
85
],
"text": [
"cyclic"
]
} |
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? | {
"answer_start": [
164
],
"text": [
"Any cyclic group with n elements"
]
} |
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? | {
"answer_start": [
264
],
"text": [
"the group of n-th complex roots of unity"
]
} |
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? | {
"answer_start": [
0
],
"text": [
"Symmetry groups"
]
} |
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? | {
"answer_start": [
93
],
"text": [
"geometric nature"
]
} |
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? | {
"answer_start": [
254
],
"text": [
"group theory"
]
} |
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? | {
"answer_start": [
532
],
"text": [
"group law"
]
} |
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? | {
"answer_start": [
724
],
"text": [
"group action"
]
} |
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? | {
"answer_start": [
10
],
"text": [
"group theory"
]
} |
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? | {
"answer_start": [
111
],
"text": [
"phase transition"
]
} |
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? | {
"answer_start": [
305
],
"text": [
"Curie temperature"
]
} |
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? | {
"answer_start": [
461
],
"text": [
"soft phonon mode"
]
} |
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? | {
"answer_start": [
35
],
"text": [
"Mathieu groups"
]
} |
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? | {
"answer_start": [
62
],
"text": [
"coding theory"
]
} |
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? | {
"answer_start": [
185
],
"text": [
"differential Galois theory"
]
} |
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? | {
"answer_start": [
487
],
"text": [
"invariant theory"
]
} |
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? | {
"answer_start": [
0
],
"text": [
"Matrix groups"
]
} |
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? | {
"answer_start": [
71
],
"text": [
"The general linear group"
]
} |
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? | {
"answer_start": [
200
],
"text": [
"matrix groups"
]
} |
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? | {
"answer_start": [
355
],
"text": [
"the special orthogonal group SO(n)"
]
} |
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? | {
"answer_start": [
448
],
"text": [
"Euler angles"
]
} |
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? | {
"answer_start": [
233
],
"text": [
"degree 5 and higher"
]
} |
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? | {
"answer_start": [
254
],
"text": [
"Abstract properties of Galois groups"
]
} |
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? | {
"answer_start": [
431
],
"text": [
"radicals"
]
} |
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? | {
"answer_start": [
35
],
"text": [
"a finite number of elements"
]
} |
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? | {
"answer_start": [
97
],
"text": [
"the order of the group"
]
} |
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? | {
"answer_start": [
439
],
"text": [
"fundamental"
]
} |
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? | {
"answer_start": [
636
],
"text": [
"S3"
]
} |
57283ac63acd2414000df75f | Group_(mathematics) | Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be sh... | What level of finality do mathematicians try to reach with math concepts? | {
"answer_start": [
34
],
"text": [
"complete classification"
]
} |
57283ac63acd2414000df760 | Group_(mathematics) | Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be sh... | What area of classification leads to troublesome arithmetic? | {
"answer_start": [
112
],
"text": [
"finite groups"
]
} |
57283ac63acd2414000df761 | Group_(mathematics) | Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be sh... | What defines finite groups of order p, a prime number, as being necessarily cyclic (abelian) groups Zp? | {
"answer_start": [
181
],
"text": [
"Lagrange's theorem"
]
} |
57283ac63acd2414000df762 | Group_(mathematics) | Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be sh... | What can be used to classify small groups even though there is no classification of all finite groups? | {
"answer_start": [
447
],
"text": [
"Computer algebra systems"
]
} |
57283ac63acd2414000df763 | Group_(mathematics) | Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be sh... | What describes finite simple groups as the building pieces for all finite groups? | {
"answer_start": [
748
],
"text": [
"The Jordan–Hölder theorem"
]
} |
57283bec2ca10214002da13c | Group_(mathematics) | Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, g • h, and g−1 must not vary wildly if g and h vary only little. Such groups are called topological groups, and they are the group objects ... | What group operations must occur for group law and topology to integrate well? | {
"answer_start": [
150
],
"text": [
"continuous functions"
]
} |
57283bec2ca10214002da13d | Group_(mathematics) | Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, g • h, and g−1 must not vary wildly if g and h vary only little. Such groups are called topological groups, and they are the group objects ... | What is the most simple example of topological groups? | {
"answer_start": [
391
],
"text": [
"reals R under addition, (R ∖ {0}, ·),"
]
} |
57283bec2ca10214002da13e | Group_(mathematics) | Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, g • h, and g−1 must not vary wildly if g and h vary only little. Such groups are called topological groups, and they are the group objects ... | What variables do locally compact groups share that can be studied by harmonic analysis? | {
"answer_start": [
577
],
"text": [
"Haar measures"
]
} |
57283d5cff5b5019007d9fb4 | Group_(mathematics) | for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is c... | What concepts are fundamental to number theory? | {
"answer_start": [
82
],
"text": [
"adele rings and adelic algebraic groups"
]
} |
57283d5cff5b5019007d9fb5 | Group_(mathematics) | for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is c... | What group uses infinite field extensions with topology? | {
"answer_start": [
208
],
"text": [
"the absolute Galois group"
]
} |
57283d5cff5b5019007d9fb6 | Group_(mathematics) | for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is c... | What is used to generalize the connection of fields and groups to infinite field extensions? | {
"answer_start": [
286
],
"text": [
"Krull topology"
]
} |
57283d5cff5b5019007d9fb7 | Group_(mathematics) | for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is c... | What group is an advanced observation of infinite field extensions and groups that is adapted for the needs of algebraic geometry? | {
"answer_start": [
510
],
"text": [
"the étale fundamental group"
]
} |
57283e51ff5b5019007d9fd8 | Group_(mathematics) | Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry o... | What concept is of basic importance in modern physics? | {
"answer_start": [
0
],
"text": [
"Lie groups"
]
} |
57283e51ff5b5019007d9fd9 | Group_(mathematics) | Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry o... | What connects continuous symmetries to conserved quantities? | {
"answer_start": [
60
],
"text": [
"Noether's theorem"
]
} |
57283e51ff5b5019007d9fda | Group_(mathematics) | Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry o... | What term describes the basic symmetries of the laws of mechanics? | {
"answer_start": [
131
],
"text": [
"Rotation"
]
} |
57283e51ff5b5019007d9fdb | Group_(mathematics) | Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry o... | What concept relates measurements of time and velocity of two observers in motion relative to each other? | {
"answer_start": [
483
],
"text": [
"Lorentz transformations"
]
} |
57283e51ff5b5019007d9fdc | Group_(mathematics) | Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry o... | What describe the complete symmetry group of Minkowski space including translations? | {
"answer_start": [
937
],
"text": [
"Poincaré group"
]
} |
57283f6d2ca10214002da17a | Group_(mathematics) | In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the non... | What product is created if the requirement that every element has an inverse is eliminated? | {
"answer_start": [
233
],
"text": [
"monoid"
]
} |
57283f6d2ca10214002da17b | Group_(mathematics) | In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the non... | What type of numbers under addition form a monoid? | {
"answer_start": [
245
],
"text": [
"natural numbers N (including 0)"
]
} |
57283f6d2ca10214002da17c | Group_(mathematics) | In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the non... | What method formally adds inverses to elements to any monoid? | {
"answer_start": [
541
],
"text": [
"the Grothendieck group"
]
} |
57283f6d2ca10214002da17d | Group_(mathematics) | In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the non... | What can be replaced to simplify abstract algebra concepts? | {
"answer_start": [
896
],
"text": [
"the binary operation"
]
} |
5728082c3acd2414000df2c1 | Central_African_Republic | The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic... | Where is the Central African Republic located? | {
"answer_start": [
195
],
"text": [
"Central Africa"
]
} |
5728082c3acd2414000df2c2 | Central_African_Republic | The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic... | What country borders CAR in the North? | {
"answer_start": [
229
],
"text": [
"Chad"
]
} |
5728082c3acd2414000df2c3 | Central_African_Republic | The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic... | What Country borders CAR in the South? | {
"answer_start": [
338
],
"text": [
"the Republic of the Congo"
]
} |
5728082c3acd2414000df2c4 | Central_African_Republic | The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic... | What Country borders CAR in the West? | {
"answer_start": [
381
],
"text": [
"Cameroon"
]
} |
5728082c3acd2414000df2c5 | Central_African_Republic | The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic... | What is the population of CAR approximately? | {
"answer_start": [
523
],
"text": [
"4.7 million"
]
} |
572808ff4b864d19001642a4 | Central_African_Republic | What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France, which ruled the country as a colony starting in the late 19th century. After gaining independence from France in 1960, the Central African Republic was ruled by a series of aut... | Who established CAR's borders? | {
"answer_start": [
136
],
"text": [
"France"
]
} |
572808ff4b864d19001642a5 | Central_African_Republic | What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France, which ruled the country as a colony starting in the late 19th century. After gaining independence from France in 1960, the Central African Republic was ruled by a series of aut... | What country ruled CAR as a colony? | {
"answer_start": [
247
],
"text": [
"France"
]
} |
572808ff4b864d19001642a6 | Central_African_Republic | What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France, which ruled the country as a colony starting in the late 19th century. After gaining independence from France in 1960, the Central African Republic was ruled by a series of aut... | When did CAR become independent? | {
"answer_start": [
254
],
"text": [
"in 1960"
]
} |
572808ff4b864d19001642a7 | Central_African_Republic | What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France, which ruled the country as a colony starting in the late 19th century. After gaining independence from France in 1960, the Central African Republic was ruled by a series of aut... | Who was the first democratically elected President? | {
"answer_start": [
430
],
"text": [
"Ange-Félix Patassé"
]
} |
572808ff4b864d19001642a8 | Central_African_Republic | What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France, which ruled the country as a colony starting in the late 19th century. After gaining independence from France in 1960, the Central African Republic was ruled by a series of aut... | What religion is a minority in CAR? | {
"answer_start": [
754
],
"text": [
"Muslim minority"
]
} |
572809fd2ca10214002d9c4e | Central_African_Republic | Approximately 10,000 years ago, desertification forced hunter-gatherer societies south into the Sahel regions of northern Central Africa, where some groups settled and began farming as part of the Neolithic Revolution. Initial farming of white yam progressed into millet and sorghum, and before 3000 BC the domestication... | How long ago did people start living in the area? | {
"answer_start": [
0
],
"text": [
"Approximately 10,000 years ago"
]
} |
572809fd2ca10214002d9c4f | Central_African_Republic | Approximately 10,000 years ago, desertification forced hunter-gatherer societies south into the Sahel regions of northern Central Africa, where some groups settled and began farming as part of the Neolithic Revolution. Initial farming of white yam progressed into millet and sorghum, and before 3000 BC the domestication... | What agricultural job did people settling in the area perform? | {
"answer_start": [
174
],
"text": [
"farming"
]
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.