• In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples...
    22 KB (3,213 words) - 17:25, 20 December 2024
  • Thumbnail for Alternating group
    alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree...
    17 KB (1,539 words) - 05:01, 21 October 2024
  • Thumbnail for Symmetric group
    outer automorphism of S6—see Automorphisms of the symmetric and alternating groups for details. Note that while A6 and A7 have an exceptional Schur multiplier...
    46 KB (6,212 words) - 00:39, 20 June 2025
  • In the mathematical area of group theory, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective...
    16 KB (1,857 words) - 04:41, 25 May 2025
  • the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out(G). If Out(G) is trivial and G has a trivial center...
    11 KB (1,123 words) - 23:09, 7 April 2025
  • and g is the order of the group of "graph automorphisms" (coming from automorphisms of the Dynkin diagram). The outer automorphism group is often, but...
    46 KB (1,789 words) - 11:28, 3 August 2024
  • Thumbnail for Exceptional object
    simple groups, the only example is in the automorphisms of the symmetric and alternating groups: for n ≥ 3 , n ≠ 6 {\displaystyle n\geq 3,n\neq 6} the alternating...
    25 KB (3,088 words) - 15:48, 11 November 2024
  • are subrepresentations of the second tensor power. In the language of modules over the group ring, the symmetric and alternating squares are C [ G ] {\displaystyle...
    16 KB (2,941 words) - 05:49, 19 May 2025
  • Thumbnail for P-group
    pi+1. The automorphism groups of p-groups are well studied. Just as every finite p-group has a non-trivial center so that the inner automorphism group is...
    21 KB (2,765 words) - 16:44, 24 May 2025
  • Thumbnail for Group of Lie type
    groups have diagram automorphisms induced by automorphisms of their Dynkin diagrams, and field automorphisms induced by automorphisms of a finite field. Analogously...
    22 KB (2,985 words) - 04:28, 23 November 2024
  • Thumbnail for Classification of finite simple groups
    or alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six exceptions, called sporadic (the Tits...
    44 KB (3,961 words) - 21:20, 25 June 2025
  • Thumbnail for Dihedral group
    18 inner automorphisms. As 2D isometry group D9, the group has mirrors at 20° intervals. The 18 inner automorphisms provide rotation of the mirrors by...
    28 KB (3,499 words) - 03:13, 9 June 2025
  • Thumbnail for Mathieu group
    alternating groups (Cameron 1992, p. 139). The only 4-transitive groups are the symmetric groups Sk for k at least 4, the alternating groups Ak for k at...
    23 KB (2,227 words) - 02:52, 3 July 2025
  • Thumbnail for Projective linear group
    Cr(Pn(k)) of birational automorphisms; any biregular automorphism is linear, so PGL coincides with the group of biregular automorphisms. Projective transformation...
    44 KB (5,613 words) - 10:17, 14 May 2025
  • Thumbnail for Elementary abelian group
    must be a prime number, and the elementary abelian groups in which the common order is p are a particular kind of p-group. A group for which p = 2 (that...
    8 KB (991 words) - 12:15, 19 May 2025
  • Thumbnail for Simple group
    {\displaystyle A_{n}} – alternating group for n ≥ 5 {\displaystyle n\geq 5} The alternating groups may be considered as groups of Lie type over the field with one...
    16 KB (2,136 words) - 01:30, 1 July 2025
  • Thumbnail for Group action
    the orbit-stabilizer theorem to count the automorphisms of a graph. Consider the cubical graph as pictured, and let G denote its automorphism group....
    46 KB (5,742 words) - 17:46, 24 May 2025
  • Thumbnail for Group homomorphism
    isomorphism. The set of all automorphisms of a group G, with functional composition as operation, itself forms a group, the automorphism group of G. It is...
    10 KB (1,538 words) - 18:09, 3 March 2025
  • Thumbnail for Unitary group
    and the field automorphism of the extension C/R (namely complex conjugation). Both these automorphisms are automorphisms of the algebraic group, have...
    21 KB (3,297 words) - 11:34, 30 April 2025
  • Thumbnail for Finite group
    considered in mathematics, after cyclic, symmetric and alternating groups, with the projective special linear groups over prime finite fields, PSL(2, p) being...
    15 KB (1,831 words) - 16:54, 2 February 2025
  • Alternating group Automorphisms of the symmetric and alternating groups Block (permutation group theory) Cayley's theorem Cycle index Frobenius group...
    4 KB (282 words) - 11:52, 17 July 2024
  • Thumbnail for Permutation group
    subset of a symmetric group that is closed under composition of permutations, contains the identity permutation, and contains the inverse permutation of each...
    23 KB (3,367 words) - 08:36, 30 June 2025
  • also the case for the symmetric and alternating square. However, we need a Haar measure on the direct product of compact groups in order to extend the theorem...
    105 KB (21,294 words) - 10:21, 1 April 2025
  • Thumbnail for Braid group
    under the symmetric group, and Y {\displaystyle Y} is the quotient by the symmetric group of the non-excluded n {\displaystyle n} -tuples. Under the dimension...
    36 KB (4,891 words) - 22:52, 19 June 2025
  • Thumbnail for Reductive group
    understood. The classification of finite simple groups says that most finite simple groups arise as the group G(k) of k-rational points of a simple algebraic...
    56 KB (8,018 words) - 09:30, 15 April 2025
  • Thumbnail for Group theory
    matrix groups conceptually similar to permutation groups, and the geometry of the action may be usefully exploited to establish properties of the group G....
    39 KB (5,086 words) - 11:47, 19 June 2025
  • Thumbnail for Janko group J2
    of index two of the group of automorphisms of the Hall–Janko graph, leading to a permutation representation of degree 100. It is also a subgroup of index...
    11 KB (909 words) - 07:06, 30 January 2025
  • Thumbnail for Cayley graph
    Cayley graph (category Permutation groups)
    graph automorphisms than the left multiplication maps, for example group automorphisms of G {\displaystyle G} which permute S {\displaystyle S} . The Cayley...
    29 KB (4,692 words) - 21:36, 19 June 2025
  • Thumbnail for Quotient group
    elements of the subgroup. In category theory, quotient groups are examples of quotient objects, which are dual to subobjects. Given a group G {\displaystyle...
    20 KB (3,753 words) - 13:10, 26 June 2025
  • Thumbnail for General linear group
    {\displaystyle \operatorname {GL} (n,F)} and its subgroups are often called linear groups or matrix groups (the automorphism group GL ⁡ ( V ) {\displaystyle \operatorname...
    24 KB (3,929 words) - 19:07, 8 May 2025