an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group. Automorphism groups...

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...

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...

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...

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...

{\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...

X admits infinitely many conformal automorphisms (in fact, the conformal automorphism group is a complex Lie group of dimension three for a sphere and...

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...

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...

"field automorphisms" (generated by a Frobenius automorphism), and g is the order of the group of "graph automorphisms" (coming from automorphisms of the...

the automorphism group of An is the symmetric group Sn, with inner automorphism group An and outer automorphism group Z2; the outer automorphism comes...

example, the group associated to G2 is the automorphism group of the octonions, and the group associated to F4 is the automorphism group of a certain...

another automorphism, and the set of automorphisms of a given graph, under the composition operation, forms a group, the automorphism group of the graph...

outer automorphisms, which are all conjugate by an inner automorphism). The automorphism group of Dn is isomorphic to the holomorph of Z {\displaystyle...

of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of S6, the symmetric group on 6 elements...

(n,p)} is the outer automorphism group of the group Z p n {\displaystyle \mathbb {Z} _{p}^{n}} , and also the automorphism group, because Z p n {\displaystyle...

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...

+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....

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}...

Similar to how the algebraic group G2 is the automorphism group of the octonions and the algebraic group F4 is the automorphism group of an Albert algebra, an...

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...

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...

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...

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...

whether the monster actually existed. Griess constructed M as the automorphism group of the Griess algebra, a 196883 dimensional commutative nonassociative...

quaternions H {\displaystyle \mathbb {H} } together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian...

linear group of dimension 6 over the finite field with 3 elements. The outer automorphism group has order 2, and the full automorphism group M12.2 is...

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...

X determines the automorphism group of X up to commensurability. Namely, let the Weyl group W be the subgroup of the orthogonal group O(Pic(X)) generated...

inner automorphism groups are never non-trivial cyclic). All non-abelian simple groups are quasisimple. The subnormal quasisimple subgroups of a group control...