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

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...

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...

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

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...

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....

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...

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

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

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

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...

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

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...

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...

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

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...

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...

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...

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...

of automorphisms of g {\displaystyle {\mathfrak {g}}} are denoted Aut ( g ) {\displaystyle {\text{Aut}}({\mathfrak {g}})} , the automorphism group of...

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...

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

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}...

+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....

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...

whether the monster actually existed. Griess constructed M as the automorphism group of the Griess algebra, a 196,884-dimensional commutative nonassociative...

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...

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...

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...