id stringlengths 24 24 | question stringlengths 1 270 | answer stringlengths 1 239 | documents listlengths 1 1 |
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57280d1c4b864d190016431b | What gives another symmetry function? | Composing two of these symmetry functions | [
"Group_(mathematics)\n\nThese 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... |
57280d1c4b864d190016431c | What group includes the symmetries of degree 4 and denoted D4? | the dihedral group | [
"Group_(mathematics)\n\nThese 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... |
57280ee3ff5b5019007d9c00 | What modern concept was created from many fields of mathematics? | abstract group | [
"Group_(mathematics)\n\nThe 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 R... |
57280ee3ff5b5019007d9c01 | The journey for answers to polynomial equations of degree higher than 4 was the original motivation for what theory? | group theory | [
"Group_(mathematics)\n\nThe 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 R... |
57280ee3ff5b5019007d9c02 | Which French mathematician expanded on earlier work of Paolo Ruffini and Joseph-Louis Lagrange? | Évariste Galois | [
"Group_(mathematics)\n\nThe 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 R... |
57280ee3ff5b5019007d9c03 | Who developed a theory giving the first abstract definition of a finite group? | Arthur Cayley | [
"Group_(mathematics)\n\nThe 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 R... |
5728106fff5b5019007d9c3a | When did the uniform theory of groups develop from different sources? | 1870 | [
"Group_(mathematics)\n\nThe 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 ... |
5728106fff5b5019007d9c3b | Who presented a method for specifying a group by means of generators and relations? | Walther von Dyck | [
"Group_(mathematics)\n\nThe 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 ... |
5728106fff5b5019007d9c3c | What theory did Hermann Weyl study in addition to locally compact groups? | The theory of Lie groups | [
"Group_(mathematics)\n\nThe 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 ... |
5728106fff5b5019007d9c3d | Who initially molded the theory of algebraic groups? | Claude Chevalley | [
"Group_(mathematics)\n\nThe 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 ... |
572811cc2ca10214002d9d22 | Where did the group of theorists first meet? | The University of Chicago | [
"Group_(mathematics)\n\nThe 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 ... |
572811cc2ca10214002d9d23 | What university class year belonged to the group of theorists? | 1960–61 | [
"Group_(mathematics)\n\nThe 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 ... |
572811cc2ca10214002d9d24 | What did the group of theorists classify in 1982? | all finite simple groups | [
"Group_(mathematics)\n\nThe 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 ... |
572811cc2ca10214002d9d25 | What two factors describe the large size of the project? | length of proof and number of researchers. | [
"Group_(mathematics)\n\nThe 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 ... |
572812cd2ca10214002d9d48 | What ideas are used to understand groups beyond symbols? | structural concepts | [
"Group_(mathematics)\n\nTo 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\",... |
572812cd2ca10214002d9d49 | What has to be compatible with the group operation? | constructions related to groups | [
"Group_(mathematics)\n\nTo 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\",... |
572812cd2ca10214002d9d4a | What concept describes groups that can be related to each other via functions? | group homomorphisms | [
"Group_(mathematics)\n\nTo 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\",... |
57281407ff5b5019007d9ca6 | What are two groups called if they include homomorphisms? | isomorphic | [
"Group_(mathematics)\n\nTwo 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... |
57281407ff5b5019007d9ca7 | What similar element do isomorphic groups carry? | isomorphic groups | [
"Group_(mathematics)\n\nTwo 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... |
57281407ff5b5019007d9ca8 | How can showing the second equality yields the first prove the concept of isomorphic groups? | applying a to the first equality yields the second | [
"Group_(mathematics)\n\nTwo 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... |
572820982ca10214002d9e86 | What is composed of two rotations? | a rotation | [
"Group_(mathematics)\n\nIn 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° f... |
572820982ca10214002d9e87 | What rotation can a rotation be reversed by? | inverse | [
"Group_(mathematics)\n\nIn 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° f... |
572820982ca10214002d9e88 | What essential condition must be met for a subset of a group to be a subgroup? | The subgroup test | [
"Group_(mathematics)\n\nIn 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° f... |
572821b7ff5b5019007d9dac | When is it preferable to consider two group elements to be the same? | irrelevant to the question whether a reflection has been performed | [
"Group_(mathematics)\n\nIn 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 r... |
572821b7ff5b5019007d9dad | What should not be considered when asking if a reflection has been performed? | rotation operations | [
"Group_(mathematics)\n\nIn 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 r... |
572821b7ff5b5019007d9dae | What number sets are used to show how subgroups can be seen as translations of the larger group? | Cosets | [
"Group_(mathematics)\n\nIn 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 r... |
572823544b864d1900164546 | What is another term for coset multiplication? | coset addition | [
"Group_(mathematics)\n\nThis 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... |
572823544b864d1900164547 | What group operation is taken by the set from the original group? | coset multiplication | [
"Group_(mathematics)\n\nThis 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... |
572823544b864d1900164548 | What group does the inverse of gN include? | quotient group | [
"Group_(mathematics)\n\nThis 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... |
572823f82ca10214002d9ec6 | What groups can be combined to describe every group? | is the quotient of the free group over the generators of the group | [
"Group_(mathematics)\n\nQuotient 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... |
572823f82ca10214002d9ec7 | What group can be the quotient of the free group over the generators of the group? | any group | [
"Group_(mathematics)\n\nQuotient 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... |
572823f82ca10214002d9ec8 | What is a finite composition of two symmetries or their inverses? | every symmetry of the square | [
"Group_(mathematics)\n\nQuotient 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... |
5728250a4b864d1900164588 | What map shows the relation between sub and quotient groups? | injective map | [
"Group_(mathematics)\n\nSub- 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 a... |
5728250a4b864d1900164589 | What are the opposites of injective maps? | surjective maps | [
"Group_(mathematics)\n\nSub- 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 a... |
5728250a4b864d190016458a | What is an example of a surjective map? | canonical map | [
"Group_(mathematics)\n\nSub- 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 a... |
5728250a4b864d190016458b | What theory address the phenomenon of homomorphisms being neither injective nor surjective? | the first isomorphism theorem | [
"Group_(mathematics)\n\nSub- 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 a... |
572826002ca10214002d9f12 | What are usually analyzed by associating groups to them and studying the elements of the corresponding groups? | Mathematical objects | [
"Group_(mathematics)\n\nGroups 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 gr... |
572826002ca10214002d9f13 | Who founded algebraic topology? | Henri Poincaré | [
"Group_(mathematics)\n\nGroups 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 gr... |
572826002ca10214002d9f14 | What did Henri Poincaré introduce when he established algebraic topology? | the fundamental group | [
"Group_(mathematics)\n\nGroups 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 gr... |
572826002ca10214002d9f15 | What topological properties translate into properties of groups? | proximity and continuity | [
"Group_(mathematics)\n\nGroups 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 gr... |
572826002ca10214002d9f16 | What represents elements of the fundamental group? | loops | [
"Group_(mathematics)\n\nGroups 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 gr... |
572826e63acd2414000df5a7 | What positive integer is used to divide the sum of two positive integers in modular mathematics? | the modulus | [
"Group_(mathematics)\n\nIn 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 a... |
572826e63acd2414000df5a8 | What results from modular addition? | the remainder of that division | [
"Group_(mathematics)\n\nIn 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 a... |
572826e63acd2414000df5a9 | What type of device can be use to demonstrate modular addition? | a clock | [
"Group_(mathematics)\n\nIn 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 a... |
572827c73acd2414000df5b5 | What term describes the group of integers related to a prime number? | modulo p | [
"Group_(mathematics)\n\nFor 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... |
572827c73acd2414000df5b6 | What integers are included in modulo p? | 1 to p − 1 | [
"Group_(mathematics)\n\nFor 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... |
572827c83acd2414000df5b7 | How many group elements exist if p=5? | four group elements | [
"Group_(mathematics)\n\nFor 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... |
572829024b864d1900164612 | What is a group labeled when the element 1 is primitive? | cyclic | [
"Group_(mathematics)\n\nIn 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 ro... |
572829024b864d1900164613 | What group is isomorphic to cyclic groups? | Any cyclic group with n elements | [
"Group_(mathematics)\n\nIn 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 ro... |
572829024b864d1900164614 | What example of cyclic group satisfies the express of zn = 1? | the group of n-th complex roots of unity | [
"Group_(mathematics)\n\nIn 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 ro... |
57282a4b2ca10214002d9fc0 | What are groups consisting of symmetries of given arithmetic concepts? | Symmetry groups | [
"Group_(mathematics)\n\nSymmetry 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 ... |
57282a4b2ca10214002d9fc1 | What term describes the introductory symmetry group of the square? | geometric nature | [
"Group_(mathematics)\n\nSymmetry 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 ... |
57282a4b2ca10214002d9fc2 | What can regarded as the study of symmetry? | group theory | [
"Group_(mathematics)\n\nSymmetry 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 ... |
57282a4b2ca10214002d9fc3 | What is the name of the rule that must be met for a group operation to occur? | group law | [
"Group_(mathematics)\n\nSymmetry 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 ... |
57282a4b2ca10214002d9fc4 | The group pattern is connected to the structure of the target by what behavior? | group action | [
"Group_(mathematics)\n\nSymmetry 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 ... |
57282d592ca10214002d9ff0 | What aids in predicting changes of physical traits? | group theory | [
"Group_(mathematics)\n\nLikewise, 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 o... |
57282d592ca10214002d9ff1 | What stage of a physical transformation can group theory be used to make prediction? | phase transition | [
"Group_(mathematics)\n\nLikewise, 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 o... |
57282d592ca10214002d9ff2 | What temperature causes the change of ferroelectric materials? | Curie temperature | [
"Group_(mathematics)\n\nLikewise, 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 o... |
57282d592ca10214002d9ff3 | What term describes the vibrational lattice mode that turns to 0 frequency at the change? | soft phonon mode | [
"Group_(mathematics)\n\nLikewise, 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 o... |
5728362a3acd2414000df70f | What are examples of finite symmetry groups used in coding theory? | Mathieu groups | [
"Group_(mathematics)\n\nFinite 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, givi... |
5728362a3acd2414000df710 | What is used for error correction of transferred data? | coding theory | [
"Group_(mathematics)\n\nFinite 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, givi... |
5728362a3acd2414000df711 | What describes functions having antiderivatives of a prescribed form? | differential Galois theory | [
"Group_(mathematics)\n\nFinite 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, givi... |
5728362a3acd2414000df712 | What concept investigates geometric elements that stay stable under group action? | invariant theory | [
"Group_(mathematics)\n\nFinite 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, givi... |
5728372a4b864d190016475c | What groups combine matrices with matrix multiplication? | Matrix groups | [
"Group_(mathematics)\n\nMatrix 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 ... |
5728372a4b864d190016475d | What includes all invertible n-by-n matrices with real entries? | The general linear group | [
"Group_(mathematics)\n\nMatrix 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 ... |
5728372a4b864d190016475e | What term describes subgroups of the general linear group? | matrix groups | [
"Group_(mathematics)\n\nMatrix 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 ... |
5728372a4b864d190016475f | What matrix group portrays all possible rotations in n dimensions? | the special orthogonal group SO(n) | [
"Group_(mathematics)\n\nMatrix 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 ... |
5728372a4b864d1900164760 | Rotation matrix groups are utilized in computer graphics with what concept? | Euler angles | [
"Group_(mathematics)\n\nMatrix 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 ... |
572838983acd2414000df749 | What degree does not include simple formulas for cubic and quatric equations? | degree 5 and higher | [
"Group_(mathematics)\n\nExchanging \"+\" 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 grou... |
572838983acd2414000df74a | What concept is associated with the solvability of polynomials? | Abstract properties of Galois groups | [
"Group_(mathematics)\n\nExchanging \"+\" 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 grou... |
572838983acd2414000df74b | What are used to express the solutions of polynomials? | radicals | [
"Group_(mathematics)\n\nExchanging \"+\" 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 grou... |
5728396e2ca10214002da110 | What does a finite group include? | a finite number of elements | [
"Group_(mathematics)\n\nA 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 pos... |
5728396e2ca10214002da111 | What is the number of elements in a group named? | the order of the group | [
"Group_(mathematics)\n\nA 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 pos... |
5728396e2ca10214002da112 | What type of class has a finite group that can be expressed as a subgroup of a symmetric group? | fundamental | [
"Group_(mathematics)\n\nA 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 pos... |
5728396e2ca10214002da113 | What can be described as the group of symmetries of an equilateral triangle? | S3 | [
"Group_(mathematics)\n\nA 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 pos... |
57283ac63acd2414000df75f | What level of finality do mathematicians try to reach with math concepts? | complete classification | [
"Group_(mathematics)\n\nMathematicians 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. Group... |
57283ac63acd2414000df760 | What area of classification leads to troublesome arithmetic? | finite groups | [
"Group_(mathematics)\n\nMathematicians 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. Group... |
57283ac63acd2414000df761 | What defines finite groups of order p, a prime number, as being necessarily cyclic (abelian) groups Zp? | Lagrange's theorem | [
"Group_(mathematics)\n\nMathematicians 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. Group... |
57283ac63acd2414000df762 | What can be used to classify small groups even though there is no classification of all finite groups? | Computer algebra systems | [
"Group_(mathematics)\n\nMathematicians 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. Group... |
57283ac63acd2414000df763 | What describes finite simple groups as the building pieces for all finite groups? | The Jordan–Hölder theorem | [
"Group_(mathematics)\n\nMathematicians 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. Group... |
57283bec2ca10214002da13c | What group operations must occur for group law and topology to integrate well? | continuous functions | [
"Group_(mathematics)\n\nSome 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... |
57283bec2ca10214002da13d | What is the most simple example of topological groups? | reals R under addition, (R ∖ {0}, ·), | [
"Group_(mathematics)\n\nSome 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... |
57283bec2ca10214002da13e | What variables do locally compact groups share that can be studied by harmonic analysis? | Haar measures | [
"Group_(mathematics)\n\nSome 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... |
57283d5cff5b5019007d9fb4 | What concepts are fundamental to number theory? | adele rings and adelic algebraic groups | [
"Group_(mathematics)\n\nfor 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 ... |
57283d5cff5b5019007d9fb5 | What group uses infinite field extensions with topology? | the absolute Galois group | [
"Group_(mathematics)\n\nfor 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 ... |
57283d5cff5b5019007d9fb6 | What is used to generalize the connection of fields and groups to infinite field extensions? | Krull topology | [
"Group_(mathematics)\n\nfor 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 ... |
57283d5cff5b5019007d9fb7 | What group is an advanced observation of infinite field extensions and groups that is adapted for the needs of algebraic geometry? | the étale fundamental group | [
"Group_(mathematics)\n\nfor 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 ... |
57283e51ff5b5019007d9fd8 | What concept is of basic importance in modern physics? | Lie groups | [
"Group_(mathematics)\n\nLie 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—imp... |
57283e51ff5b5019007d9fd9 | What connects continuous symmetries to conserved quantities? | Noether's theorem | [
"Group_(mathematics)\n\nLie 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—imp... |
57283e51ff5b5019007d9fda | What term describes the basic symmetries of the laws of mechanics? | Rotation | [
"Group_(mathematics)\n\nLie 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—imp... |
57283e51ff5b5019007d9fdb | What concept relates measurements of time and velocity of two observers in motion relative to each other? | Lorentz transformations | [
"Group_(mathematics)\n\nLie 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—imp... |
57283e51ff5b5019007d9fdc | What describe the complete symmetry group of Minkowski space including translations? | Poincaré group | [
"Group_(mathematics)\n\nLie 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—imp... |
57283f6d2ca10214002da17a | What product is created if the requirement that every element has an inverse is eliminated? | monoid | [
"Group_(mathematics)\n\nIn 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 ... |
57283f6d2ca10214002da17b | What type of numbers under addition form a monoid? | natural numbers N (including 0) | [
"Group_(mathematics)\n\nIn 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 ... |
57283f6d2ca10214002da17c | What method formally adds inverses to elements to any monoid? | the Grothendieck group | [
"Group_(mathematics)\n\nIn 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 ... |
57283f6d2ca10214002da17d | What can be replaced to simplify abstract algebra concepts? | the binary operation | [
"Group_(mathematics)\n\nIn 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 ... |
5728082c3acd2414000df2c1 | Where is the Central African Republic located? | Central Africa | [
"Central_African_Republic\n\nThe 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 ... |
5728082c3acd2414000df2c2 | What country borders CAR in the North? | Chad | [
"Central_African_Republic\n\nThe 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 ... |
5728082c3acd2414000df2c3 | What Country borders CAR in the South? | the Republic of the Congo | [
"Central_African_Republic\n\nThe 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 ... |
5728082c3acd2414000df2c4 | What Country borders CAR in the West? | Cameroon | [
"Central_African_Republic\n\nThe 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 ... |
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