• an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group. Automorphism groups...
    11 KB (1,687 words) - 12:56, 15 May 2023
  • In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the...
    11 KB (1,689 words) - 22:20, 13 October 2024
  • mathematics, the outer automorphism group of a group, G, is the quotient, Aut(G) / Inn(G), where Aut(G) is the automorphism group of G and Inn(G) is the...
    11 KB (1,091 words) - 04:25, 5 March 2024
  • Thumbnail for Automorphism
    The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object. In...
    11 KB (1,337 words) - 08:52, 12 September 2024
  • Thumbnail for Dihedral group
    outer automorphisms, which are all conjugate by an inner automorphism). The automorphism group of Dn is isomorphic to the holomorph of Z {\displaystyle...
    27 KB (3,379 words) - 04:52, 19 September 2024
  • {\displaystyle \mathbb {Z} _{7}} by 3, modulo 7, is an automorphism of order 6 in the automorphism group, because 3 6 ≡ 1 ( mod 7 ) , {\displaystyle 3^{6}\equiv...
    12 KB (2,043 words) - 06:03, 31 August 2024
  • Thumbnail for General linear group
    is the outer automorphism group of the group Zpn, and also the automorphism group, because Zpn is abelian, so the inner automorphism group is trivial....
    23 KB (2,965 words) - 00:14, 1 September 2024
  • Thumbnail for Symmetric group
    is the full automorphism group of An: Aut(An) ≅ Sn. Conjugation by even elements are inner automorphisms of An while the outer automorphism of An of order...
    46 KB (6,130 words) - 06:34, 24 May 2024
  • Thumbnail for Alternating group
    the automorphism group of An is the symmetric group Sn, with inner automorphism group An and outer automorphism group Z2; the outer automorphism comes...
    17 KB (1,539 words) - 05:01, 21 October 2024
  • of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of S6, the symmetric group on 6 elements...
    22 KB (3,212 words) - 00:31, 20 June 2024
  • "field automorphisms" (generated by a Frobenius automorphism), and g is the order of the group of "graph automorphisms" (coming from automorphisms of the...
    46 KB (1,789 words) - 11:28, 3 August 2024
  • another automorphism, and the set of automorphisms of a given graph, under the composition operation, forms a group, the automorphism group of the graph...
    14 KB (1,352 words) - 15:22, 1 August 2024
  • X admits infinitely many conformal automorphisms (in fact, the conformal automorphism group is a complex Lie group of dimension three for a sphere and...
    18 KB (2,781 words) - 13:17, 26 August 2024
  • 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 denoted...
    10 KB (1,531 words) - 12:55, 1 October 2024
  • group, G / Z(G), is isomorphic to the inner automorphism group, Inn(G). A group G is abelian if and only if Z(G) = G. At the other extreme, a group is...
    12 KB (1,184 words) - 20:06, 14 May 2024
  • mathematics, a group G is said to be complete if every automorphism of G is inner, and it is centerless; that is, it has a trivial outer automorphism group and trivial...
    4 KB (534 words) - 22:53, 5 June 2022
  • inner automorphism groups are never non-trivial cyclic). All non-abelian simple groups are quasisimple. The subnormal quasisimple subgroups of a group control...
    2 KB (296 words) - 23:43, 12 August 2023
  • Thumbnail for P-group
    quotient of the group, every finite p-group has a non-trivial outer automorphism group. Every automorphism of G induces an automorphism on G/Φ(G), where...
    21 KB (2,753 words) - 13:08, 25 October 2023
  • mathematical group theory, the automorphism group of a free group is a discrete group of automorphisms of a free group. The quotient by the inner automorphisms is...
    3 KB (368 words) - 01:55, 29 May 2024
  • Thumbnail for Mathieu group
    permutation groups, as well as automorphism groups of Steiner systems. After the Mathieu groups, no new sporadic groups were found until 1965, when the group J1...
    22 KB (2,130 words) - 14:20, 15 May 2024
  • Thumbnail for Tits group
    automorphism group has order 2, with the full automorphism group being the group 2F4(2). The Tits group occurs as a maximal subgroup of the Fischer group Fi22...
    6 KB (718 words) - 12:35, 30 October 2024
  • ασ of the group X(F) Any automorphism π of the Dynkin diagram induces an automorphism απ of the group X(F). The Steinberg and Chevalley groups can be constructed...
    17 KB (2,067 words) - 00:02, 3 December 2023
  • Thumbnail for Klein four-group
    four-group are interchangeable: the automorphism group of V is thus the group of permutations of these three elements, that is, the symmetric group S 3...
    10 KB (1,375 words) - 10:24, 31 October 2024
  • Thumbnail for Reductive group
    +x_{2n-1}x_{2n}.} The outer automorphism group of a split reductive group G over a field k is isomorphic to the automorphism group of the root datum of G....
    55 KB (7,845 words) - 18:28, 24 April 2024
  • Thumbnail for Direct product of groups
    there exists an automorphism σ of G × G that switches the two factors, i.e. σ(g1, g2) = (g2, g1). For another example, the automorphism group of Z × Z is...
    26 KB (2,932 words) - 23:03, 19 April 2024
  • Thumbnail for Monster group
    whether the monster actually existed. Griess constructed M as the automorphism group of the Griess algebra, a 196,884-dimensional commutative nonassociative...
    35 KB (2,971 words) - 13:54, 22 October 2024
  • Thumbnail for Projective linear group
    certain axioms – an automorphism of a projective space thus defined then being an automorphism f of the set of points and an automorphism g of the set of...
    44 KB (5,611 words) - 09:09, 9 September 2024
  • Thumbnail for Group of Lie type
    of An; further orthogonal groups 2Dn, from the order 2 automorphism of Dn; the new series 2E6, from the order 2 automorphism of E6; the new series 3D4...
    22 KB (2,985 words) - 10:42, 28 March 2023
  • Thumbnail for Abelian group
    theorem shows that to compute the automorphism group of G {\displaystyle G} it suffices to compute the automorphism groups of the Sylow p {\displaystyle p}...
    36 KB (5,284 words) - 17:38, 8 October 2024
  • Thumbnail for Quaternion group
    \{e,{\bar {e}}\}} . The inner automorphism group of Q8 is given by the group modulo its center, i.e. the factor group Q 8 / { e , e ¯ } , {\displaystyle...
    26 KB (3,724 words) - 19:00, 13 September 2024