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-110900 | 572807e2ff5b5019007d9b66 | Group_(mathematics) | Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie... | What groups are are utilized toward comprehending symmetry wonders in chemistry? | What groups are are utilized toward comprehending symmetry wonders in chemistry? | [
"What groups are are utilized toward comprehending symmetry wonders in chemistry? "
] | {
"text": [
"Point groups"
],
"answer_start": [
400
]
} |
gem-squad_v2-train-110901 | 572807e2ff5b5019007d9b67 | Group_(mathematics) | Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie... | What groups can express the physical symmetry hidden behind special relativity? | What groups can express the physical symmetry hidden behind special relativity? | [
"What groups can express the physical symmetry hidden behind special relativity?"
] | {
"text": [
"Poincaré groups"
],
"answer_start": [
488
]
} |
gem-squad_v2-train-110902 | 5a81b6a631013a001a334dbd | Group_(mathematics) | Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie... | What encodes a symmetry group? | What encodes a symmetry group? | [
"What encodes a symmetry group?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110903 | 5a81b6a631013a001a334dbe | Group_(mathematics) | Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie... | What are groups dissimilar to? | What are groups dissimilar to? | [
"What are groups dissimilar to?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110904 | 5a81b6a631013a001a334dbf | Group_(mathematics) | Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie... | Point groups are used in what form of physics? | Point groups are used in what form of physics? | [
"Point groups are used in what form of physics?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110905 | 5a81b6a631013a001a334dc0 | Group_(mathematics) | Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie... | Poincare groups are used to understand molecular what? | Poincare groups are used to understand molecular what? | [
"Poincare groups are used to understand molecular what?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110906 | 5a81b6a631013a001a334dc1 | Group_(mathematics) | Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie... | What do Lie groups express in terms of special relativity? | What do Lie groups express in terms of special relativity? | [
"What do Lie groups express in terms of special relativity?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110907 | 572809ab2ca10214002d9c38 | Group_(mathematics) | The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies gr... | Where did the idea of a group come from? | Where did the idea of a group come from? | [
"Where did the idea of a group come from?"
] | {
"text": [
"the study of polynomial equations,"
],
"answer_start": [
34
]
} |
gem-squad_v2-train-110908 | 572809ab2ca10214002d9c39 | Group_(mathematics) | The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies gr... | When was the group notion summed up and solidly settled? | When was the group notion summed up and solidly settled? | [
"When was the group notion summed up and solidly settled?"
] | {
"text": [
"1870"
],
"answer_start": [
250
]
} |
gem-squad_v2-train-110909 | 572809ab2ca10214002d9c3a | Group_(mathematics) | The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies gr... | What are smaller and easier to understand groups broken down into? | What are smaller and easier to understand groups broken down into? | [
"What are smaller and easier to understand groups broken down into?"
] | {
"text": [
"subgroups, quotient groups and simple groups."
],
"answer_start": [
480
]
} |
gem-squad_v2-train-110910 | 572809ab2ca10214002d9c3b | Group_(mathematics) | The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies gr... | When was announcement for the classification of finite simple groups? | When was announcement for the classification of finite simple groups? | [
"When was announcement for the classification of finite simple groups?"
] | {
"text": [
"1983"
],
"answer_start": [
872
]
} |
gem-squad_v2-train-110911 | 5a81b7f631013a001a334dd9 | Group_(mathematics) | The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies gr... | Who generalized group notion in 1870? | Who generalized group notion in 1870? | [
"Who generalized group notion in 1870?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110912 | 5a81b7f631013a001a334dda | Group_(mathematics) | The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies gr... | What year did mathematicians begin studying groups in their own right? | What year did mathematicians begin studying groups in their own right? | [
"What year did mathematicians begin studying groups in their own right?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110913 | 5a81b7f631013a001a334ddb | Group_(mathematics) | The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies gr... | What are the two subgroups? | What are the two subgroups? | [
"What are the two subgroups?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110914 | 5a81b7f631013a001a334ddc | Group_(mathematics) | The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies gr... | When did theorists begin exploring the theoretical and computational point of view? | When did theorists begin exploring the theoretical and computational point of view? | [
"When did theorists begin exploring the theoretical and computational point of view?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110915 | 5a81b7f631013a001a334ddd | Group_(mathematics) | The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies gr... | When did the geometric group theory become less active? | When did the geometric group theory become less active? | [
"When did the geometric group theory become less active?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110916 | 57280b474b864d19001642e6 | Group_(mathematics) | The set G is called the underlying set of the group (G, •). Often the group's underlying set G is used as a short name for the group (G, •). Along the same lines, shorthand expressions such as "a subset of the group G" or "an element of group G" are used when what is actually meant is "a subset of the underlying set G ... | What is known as underlying set of the group? | What is known as underlying set of the group? | [
"What is known as underlying set of the group?"
] | {
"text": [
"The set"
],
"answer_start": [
0
]
} |
gem-squad_v2-train-110917 | 57280b474b864d19001642e7 | Group_(mathematics) | The set G is called the underlying set of the group (G, •). Often the group's underlying set G is used as a short name for the group (G, •). Along the same lines, shorthand expressions such as "a subset of the group G" or "an element of group G" are used when what is actually meant is "a subset of the underlying set G ... | What is often utilized as a short name for the group? | What is often utilized as a short name for the group? | [
"What is often utilized as a short name for the group?"
] | {
"text": [
"the group's underlying set"
],
"answer_start": [
66
]
} |
gem-squad_v2-train-110918 | 57280b474b864d19001642e8 | Group_(mathematics) | The set G is called the underlying set of the group (G, •). Often the group's underlying set G is used as a short name for the group (G, •). Along the same lines, shorthand expressions such as "a subset of the group G" or "an element of group G" are used when what is actually meant is "a subset of the underlying set G ... | What expressions are utilized when is really implied to be a longer expression? | What expressions are utilized when is really implied to be a longer expression? | [
"What expressions are utilized when is really implied to be a longer expression?"
] | {
"text": [
"shorthand expressions"
],
"answer_start": [
163
]
} |
gem-squad_v2-train-110919 | 5a81b8ad31013a001a334de3 | Group_(mathematics) | The set G is called the underlying set of the group (G, •). Often the group's underlying set G is used as a short name for the group (G, •). Along the same lines, shorthand expressions such as "a subset of the group G" or "an element of group G" are used when what is actually meant is "a subset of the underlying set G ... | What cannot be used as the short name for the group? | What cannot be used as the short name for the group? | [
"What cannot be used as the short name for the group?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110920 | 5a81b8ad31013a001a334de4 | Group_(mathematics) | The set G is called the underlying set of the group (G, •). Often the group's underlying set G is used as a short name for the group (G, •). Along the same lines, shorthand expressions such as "a subset of the group G" or "an element of group G" are used when what is actually meant is "a subset of the underlying set G ... | What is the overlaying set of the group? | What is the overlaying set of the group? | [
"What is the overlaying set of the group?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110921 | 5a81b8ad31013a001a334de5 | Group_(mathematics) | The set G is called the underlying set of the group (G, •). Often the group's underlying set G is used as a short name for the group (G, •). Along the same lines, shorthand expressions such as "a subset of the group G" or "an element of group G" are used when what is actually meant is "a subset of the underlying set G ... | What can be unclear for the G symbol? | What can be unclear for the G symbol? | [
"What can be unclear for the G symbol?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110922 | 5a81b8ad31013a001a334de6 | Group_(mathematics) | The set G is called the underlying set of the group (G, •). Often the group's underlying set G is used as a short name for the group (G, •). Along the same lines, shorthand expressions such as "a subset of the group G" or "an element of group G" are used when what is actually meant is "a subset of the underlying set G ... | A subset of the underlying set G of the group (G) cannot be written using what kind of expression? | A subset of the underlying set G of the group (G) cannot be written using what kind of expression? | [
"A subset of the underlying set G of the group (G) cannot be written using what kind of expression?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110923 | 57280d1c4b864d190016431a | Group_(mathematics) | These symmetries are represented by functions. Each of these functions sends a point in the square to the corresponding point under the symmetry. For example, r1 sends a point to its rotation 90° clockwise around the square's center, and fh sends a point to its reflection across the square's vertical middle line. Compo... | What sends a point in the square to the relating point under the symmetry? | What sends a point in the square to the relating point under the symmetry? | [
"What sends a point in the square to the relating point under the symmetry? "
] | {
"text": [
"functions"
],
"answer_start": [
61
]
} |
gem-squad_v2-train-110924 | 57280d1c4b864d190016431b | Group_(mathematics) | These symmetries are represented by functions. Each of these functions sends a point in the square to the corresponding point under the symmetry. For example, r1 sends a point to its rotation 90° clockwise around the square's center, and fh sends a point to its reflection across the square's vertical middle line. Compo... | What gives another symmetry function? | What gives another symmetry function? | [
" What gives another symmetry function?"
] | {
"text": [
"Composing two of these symmetry functions"
],
"answer_start": [
315
]
} |
gem-squad_v2-train-110925 | 57280d1c4b864d190016431c | Group_(mathematics) | These symmetries are represented by functions. Each of these functions sends a point in the square to the corresponding point under the symmetry. For example, r1 sends a point to its rotation 90° clockwise around the square's center, and fh sends a point to its reflection across the square's vertical middle line. Compo... | What group includes the symmetries of degree 4 and denoted D4? | What group includes the symmetries of degree 4 and denoted D4? | [
"What group includes the symmetries of degree 4 and denoted D4?"
] | {
"text": [
"the dihedral group"
],
"answer_start": [
432
]
} |
gem-squad_v2-train-110926 | 5a81b9e531013a001a334df5 | Group_(mathematics) | These symmetries are represented by functions. Each of these functions sends a point in the square to the corresponding point under the symmetry. For example, r1 sends a point to its rotation 90° clockwise around the square's center, and fh sends a point to its reflection across the square's vertical middle line. Compo... | What are represented by symmetries? | What are represented by symmetries? | [
"What are represented by symmetries?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110927 | 5a81b9e531013a001a334df6 | Group_(mathematics) | These symmetries are represented by functions. Each of these functions sends a point in the square to the corresponding point under the symmetry. For example, r1 sends a point to its rotation 90° clockwise around the square's center, and fh sends a point to its reflection across the square's vertical middle line. Compo... | r1 is part of what group? | r1 is part of what group? | [
"r1 is part of what group?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110928 | 5a81b9e531013a001a334df7 | Group_(mathematics) | These symmetries are represented by functions. Each of these functions sends a point in the square to the corresponding point under the symmetry. For example, r1 sends a point to its rotation 90° clockwise around the square's center, and fh sends a point to its reflection across the square's vertical middle line. Compo... | Both symmetries are applied to what? | Both symmetries are applied to what? | [
"Both symmetries are applied to what?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110929 | 5a81b9e531013a001a334df8 | Group_(mathematics) | These symmetries are represented by functions. Each of these functions sends a point in the square to the corresponding point under the symmetry. For example, r1 sends a point to its rotation 90° clockwise around the square's center, and fh sends a point to its reflection across the square's vertical middle line. Compo... | What is written from left to right? | What is written from left to right? | [
"What is written from left to right?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110930 | 5a81b9e531013a001a334df9 | Group_(mathematics) | These symmetries are represented by functions. Each of these functions sends a point in the square to the corresponding point under the symmetry. For example, r1 sends a point to its rotation 90° clockwise around the square's center, and fh sends a point to its reflection across the square's vertical middle line. Compo... | How many degrees is fh rotated? | How many degrees is fh rotated? | [
"How many degrees is fh rotated?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110931 | 57280ee3ff5b5019007d9c00 | Group_(mathematics) | The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagr... | What modern concept was created from many fields of mathematics? | What modern concept was created from many fields of mathematics? | [
"What modern concept was created from many fields of mathematics?"
] | {
"text": [
"abstract group"
],
"answer_start": [
25
]
} |
gem-squad_v2-train-110932 | 57280ee3ff5b5019007d9c01 | Group_(mathematics) | The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagr... | The journey for answers to polynomial equations of degree higher than 4 was the original motivation for what theory? | The journey for answers to polynomial equations of degree higher than 4 was the original motivation for what theory? | [
"The journey for answers to polynomial equations of degree higher than 4 was the original motivation for what theory?"
] | {
"text": [
"group theory"
],
"answer_start": [
116
]
} |
gem-squad_v2-train-110933 | 57280ee3ff5b5019007d9c02 | Group_(mathematics) | The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagr... | Which French mathematician expanded on earlier work of Paolo Ruffini and Joseph-Louis Lagrange? | Which French mathematician expanded on earlier work of Paolo Ruffini and Joseph-Louis Lagrange? | [
" Which French mathematician expanded on earlier work of Paolo Ruffini and Joseph-Louis Lagrange?"
] | {
"text": [
"Évariste Galois"
],
"answer_start": [
244
]
} |
gem-squad_v2-train-110934 | 57280ee3ff5b5019007d9c03 | Group_(mathematics) | The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagr... | Who developed a theory giving the first abstract definition of a finite group? | Who developed a theory giving the first abstract definition of a finite group? | [
" Who developed a theory giving the first abstract definition of a finite group?"
] | {
"text": [
"Arthur Cayley"
],
"answer_start": [
725
]
} |
gem-squad_v2-train-110935 | 5a81bcfc31013a001a334e19 | Group_(mathematics) | The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagr... | People were looking for polynomial equations under what number? | People were looking for polynomial equations under what number? | [
"People were looking for polynomial equations under what number?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110936 | 5a81bcfc31013a001a334e1a | Group_(mathematics) | The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagr... | Whose work did Paolo Ruffini and Joseph-Louis Lagrange base their work on? | Whose work did Paolo Ruffini and Joseph-Louis Lagrange base their work on? | [
"Whose work did Paolo Ruffini and Joseph-Louis Lagrange base their work on?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110937 | 5a81bcfc31013a001a334e1b | Group_(mathematics) | The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagr... | What was Augustin Louis Cauchy's nationality? | What was Augustin Louis Cauchy's nationality? | [
"What was Augustin Louis Cauchy's nationality?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110938 | 5a81bcfc31013a001a334e1c | Group_(mathematics) | The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagr... | When did Arthur Cayley publish On the theory of groups? | When did Arthur Cayley publish On the theory of groups? | [
"When did Arthur Cayley publish On the theory of groups?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110939 | 5a81bcfc31013a001a334e1d | Group_(mathematics) | The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagr... | What equation did Galois create that created an abstract definition? | What equation did Galois create that created an abstract definition? | [
"What equation did Galois create that created an abstract definition?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110940 | 5728106fff5b5019007d9c3a | 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... | When did the uniform theory of groups develop from different sources? | When did the uniform theory of groups develop from different sources? | [
"When did the uniform theory of groups develop from different sources? "
] | {
"text": [
"1870"
],
"answer_start": [
158
]
} |
gem-squad_v2-train-110941 | 5728106fff5b5019007d9c3b | 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 presented a method for specifying a group by means of generators and relations? | Who presented a method for specifying a group by means of generators and relations? | [
"Who presented a method for specifying a group by means of generators and relations?"
] | {
"text": [
"Walther von Dyck"
],
"answer_start": [
165
]
} |
gem-squad_v2-train-110942 | 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? | What theory did Hermann Weyl study in addition to locally compact groups? | [
"What theory did Hermann Weyl study in addition to locally compact groups?"
] | {
"text": [
"The theory of Lie groups"
],
"answer_start": [
636
]
} |
gem-squad_v2-train-110943 | 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? | Who initially molded the theory of algebraic groups? | [
"Who initially molded the theory of algebraic groups?"
] | {
"text": [
"Claude Chevalley"
],
"answer_start": [
841
]
} |
gem-squad_v2-train-110944 | 5a81bdcf31013a001a334e23 | 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 did Walter von Dyck publish in 1870? | What did Walter von Dyck publish in 1870? | [
"What did Walter von Dyck publish in 1870?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110945 | 5a81bdcf31013a001a334e24 | 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 wrote the theory of Lie groups? | Who wrote the theory of Lie groups? | [
"Who wrote the theory of Lie groups?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110946 | 5a81bdcf31013a001a334e25 | 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 did Jacques Tits shape first? | What did Jacques Tits shape first? | [
"What did Jacques Tits shape first?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110947 | 5a81bdcf31013a001a334e26 | 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 was Issai Schur's theory on? | What was Issai Schur's theory on? | [
"What was Issai Schur's theory on?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110948 | 5a81bdcf31013a001a334e27 | 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 worked on representation theory of lie groups? | Who worked on representation theory of lie groups? | [
"Who worked on representation theory of lie groups?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110949 | 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? | Where did the group of theorists first meet? | [
"Where did the group of theorists first meet?"
] | {
"text": [
"The University of Chicago"
],
"answer_start": [
0
]
} |
gem-squad_v2-train-110950 | 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? | What university class year belonged to the group of theorists? | [
"What university class year belonged to the group of theorists?"
] | {
"text": [
"1960–61"
],
"answer_start": [
28
]
} |
gem-squad_v2-train-110951 | 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? | What did the group of theorists classify in 1982? | [
"What did the group of theorists classify in 1982?"
] | {
"text": [
"all finite simple groups"
],
"answer_start": [
253
]
} |
gem-squad_v2-train-110952 | 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? | What two factors describe the large size of the project? | [
"What two factors describe the large size of the project?"
] | {
"text": [
"length of proof and number of researchers."
],
"answer_start": [
369
]
} |
gem-squad_v2-train-110953 | 5a81be8331013a001a334e2d | 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 was classified in 1960? | What was classified in 1960? | [
"What was classified in 1960?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110954 | 5a81be8331013a001a334e2e | 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... | Who founded the University of Chicago's Group Theory Year? | Who founded the University of Chicago's Group Theory Year? | [
"Who founded the University of Chicago's Group Theory Year?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110955 | 5a81be8331013a001a334e2f | 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 no longer has much impact on other fields? | What no longer has much impact on other fields? | [
"What no longer has much impact on other fields?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110956 | 5a81be8331013a001a334e30 | 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 was shorter about this project as compared to other endeavors? | What was shorter about this project as compared to other endeavors? | [
"What was shorter about this project as compared to other endeavors?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110957 | 5a81be8331013a001a334e31 | 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... | When did the University of Chicago begin? | When did the University of Chicago begin? | [
"When did the University of Chicago begin?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110958 | 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? | What ideas are used to understand groups beyond symbols? | [
"What ideas are used to understand groups beyond symbols?"
] | {
"text": [
"structural concepts"
],
"answer_start": [
84
]
} |
gem-squad_v2-train-110959 | 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? | What has to be compatible with the group operation? | [
"What has to be compatible with the group operation?"
] | {
"text": [
"constructions related to groups"
],
"answer_start": [
305
]
} |
gem-squad_v2-train-110960 | 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? | What concept describes groups that can be related to each other via functions? | [
"What concept describes groups that can be related to each other via functions?"
] | {
"text": [
"group homomorphisms"
],
"answer_start": [
533
]
} |
gem-squad_v2-train-110961 | 5a81bf4631013a001a334e37 | 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 is needed to understand structural concepts? | What is needed to understand structural concepts? | [
"What is needed to understand structural concepts?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110962 | 5a81bf4631013a001a334e38 | 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... | Both groups and sets have what? | Both groups and sets have what? | [
"Both groups and sets have what?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110963 | 5a81bf4631013a001a334e39 | 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 function is used to break groups into pieces? | What function is used to break groups into pieces? | [
"What function is used to break groups into pieces?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110964 | 5a81bf4631013a001a334e3a | 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... | Groups do not need to respect structures in what sense? | Groups do not need to respect structures in what sense? | [
"Groups do not need to respect structures in what sense?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110965 | 5a81bf4631013a001a334e3b | 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 is an example of a subgroup? | What is an example of a subgroup? | [
"What is an example of a subgroup?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110966 | 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? | What are two groups called if they include homomorphisms? | [
"What are two groups called if they include homomorphisms?"
] | {
"text": [
"isomorphic"
],
"answer_start": [
30
]
} |
gem-squad_v2-train-110967 | 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? | What similar element do isomorphic groups carry? | [
"What similar element do isomorphic groups carry?"
] | {
"text": [
"isomorphic groups"
],
"answer_start": [
326
]
} |
gem-squad_v2-train-110968 | 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? | How can showing the second equality yields the first prove the concept of isomorphic groups? | [
"How can showing the second equality yields the first prove the concept of isomorphic groups?"
] | {
"text": [
"applying a to the first equality yields the second"
],
"answer_start": [
489
]
} |
gem-squad_v2-train-110969 | 5a81c05031013a001a334e41 | 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 no group homomorphisms are found? | What are two groups called if no group homomorphisms are found? | [
"What are two groups called if no group homomorphisms are found?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110970 | 5a81c05031013a001a334e42 | 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 ... | Isomorphic groups carry different what? | Isomorphic groups carry different what? | [
"Isomorphic groups carry different what?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110971 | 5a81c05031013a001a334e43 | 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 gives the identity of a? | What gives the identity of a? | [
"What gives the identity of a?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110972 | 5a81c05031013a001a334e44 | 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 ... | From what point of view do isomorphic groups have different information? | From what point of view do isomorphic groups have different information? | [
"From what point of view do isomorphic groups have different information?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110973 | 5a81c05031013a001a334e45 | 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 does the second equality disprove the concept of isomorphic groups? | How does the second equality disprove the concept of isomorphic groups? | [
"How does the second equality disprove the concept of isomorphic groups?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110974 | 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? | What is composed of two rotations? | [
"What is composed of two rotations?"
] | {
"text": [
"a rotation"
],
"answer_start": [
179
]
} |
gem-squad_v2-train-110975 | 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? | What rotation can a rotation be reversed by? | [
"What rotation can a rotation be reversed by?"
] | {
"text": [
"inverse"
],
"answer_start": [
232
]
} |
gem-squad_v2-train-110976 | 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? | What essential condition must be met for a subset of a group to be a subgroup? | [
"What essential condition must be met for a subset of a group to be a subgroup?"
] | {
"text": [
"The subgroup test"
],
"answer_start": [
381
]
} |
gem-squad_v2-train-110977 | 5a81c12031013a001a334e4b | 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 defined when moving in the opposite direction? | What is defined when moving in the opposite direction? | [
"What is defined when moving in the opposite direction?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110978 | 5a81c12031013a001a334e4c | 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... | 270 for 180 is an example of what kind of rotation? | 270 for 180 is an example of what kind of rotation? | [
"270 for 180 is an example of what kind of rotation?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110979 | 5a81c12031013a001a334e4d | 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... | subgroup R is made up of the inverse and what? | subgroup R is made up of the inverse and what? | [
"subgroup R is made up of the inverse and what?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110980 | 5a81c12031013a001a334e4e | 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 unimportant for understanding the group as a whole? | What is unimportant for understanding the group as a whole? | [
"What is unimportant for understanding the group as a whole?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110981 | 5a81c12031013a001a334e4f | 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... | Knowing the group as a whole is important for understanding what? | Knowing the group as a whole is important for understanding what? | [
"Knowing the group as a whole is important for understanding what?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110982 | 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? | When is it preferable to consider two group elements to be the same? | [
"When is it preferable to consider two group elements to be the same?"
] | {
"text": [
"irrelevant to the question whether a reflection has been performed"
],
"answer_start": [
338
]
} |
gem-squad_v2-train-110983 | 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? | What should not be considered when asking if a reflection has been performed? | [
"What should not be considered when asking if a reflection has been performed?"
] | {
"text": [
"rotation operations"
],
"answer_start": [
314
]
} |
gem-squad_v2-train-110984 | 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? | What number sets are used to show how subgroups can be seen as translations of the larger group? | [
"What number sets are used to show how subgroups can be seen as translations of the larger group?"
] | {
"text": [
"Cosets"
],
"answer_start": [
406
]
} |
gem-squad_v2-train-110985 | 5a81c9c731013a001a334ebd | 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 are two group elements considered different? | When are two group elements considered different? | [
"When are two group elements considered different?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110986 | 5a81c9c731013a001a334ebe | 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... | In D4 the square can return to the r2 configuration by applying only what? | In D4 the square can return to the r2 configuration by applying only what? | [
"In D4 the square can return to the r2 configuration by applying only what?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110987 | 5a81c9c731013a001a334ebf | 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 question are cosets irrelevant to? | What question are cosets irrelevant to? | [
"What question are cosets irrelevant to?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110988 | 5a81c9c731013a001a334ec0 | 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 does group element g define? | What does group element g define? | [
"What does group element g define?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110989 | 5a81c9c731013a001a334ec1 | 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 are rotation operations considered? | When are rotation operations considered? | [
"When are rotation operations considered?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110990 | 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? | What is another term for coset multiplication? | [
"What is another term for coset multiplication?"
] | {
"text": [
"coset addition"
],
"answer_start": [
79
]
} |
gem-squad_v2-train-110991 | 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? | What group operation is taken by the set from the original group? | [
"What group operation is taken by the set from the original group?"
] | {
"text": [
"coset multiplication"
],
"answer_start": [
54
]
} |
gem-squad_v2-train-110992 | 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? | What group does the inverse of gN include? | [
"What group does the inverse of gN include?"
] | {
"text": [
"quotient group"
],
"answer_start": [
523
]
} |
gem-squad_v2-train-110993 | 5a81caf731013a001a334edb | 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 the original group sometimes called? | What is the original group sometimes called? | [
"What is the original group sometimes called?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110994 | 5a81caf731013a001a334edc | 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 are group homomorphisms called? | What are group homomorphisms called? | [
"What are group homomorphisms called?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110995 | 5a81caf731013a001a334edd | 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 does the Coset gN serve as? | What does the Coset gN serve as? | [
"What does the Coset gN serve as?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110996 | 5a81caf731013a001a334ede | 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 original group from the set? | What group operation is taken by the original group from the set? | [
"What group operation is taken by the original group from the set?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110997 | 5a81caf731013a001a334edf | 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 ... | The inverse of gN excludes what group? | The inverse of gN excludes what group? | [
"The inverse of gN excludes what group?"
] | {
"text": [],
"answer_start": []
} |
gem-squad_v2-train-110998 | 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? | What groups can be combined to describe every group? | [
"What groups can be combined to describe every group?"
] | {
"text": [
"is the quotient of the free group over the generators of the group"
],
"answer_start": [
107
]
} |
gem-squad_v2-train-110999 | 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? | What group can be the quotient of the free group over the generators of the group? | [
"What group can be the quotient of the free group over the generators of the group?"
] | {
"text": [
"any group"
],
"answer_start": [
97
]
} |
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