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 outer automorphism group is written as d⋅f⋅g, where d is the order of the group of "diagonal automorphisms", f is the order of the (cyclic) group of...

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...

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

= 2, the group has a non-trivial center, while for the case n = 6, there is an outer automorphism. The automorphism group of a simple group is an almost...

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...

complex conjugation and by the outer automorphism which already exists as a complex automorphism. As well as the complex Lie group of type E6, there are five...

exceptional outer automorphism of An so Sn is not the full automorphism group of An. Conversely, for n ≠ 6, Sn has no outer automorphisms, and for n ≠...

mathematics, Out(Fn) is the outer automorphism group of a free group on n generators. These groups are at universal stage in geometric group theory, as they act...

Schur multiplier of E8(q) is trivial, and its outer automorphism group is that of field automorphisms (i.e., cyclic of order f if q = pf, where p is...

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

monster, and the Conway group Co1. The Schur multiplier and the outer automorphism group of the monster are both trivial. The minimal degree of a faithful...

+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...

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...

Non-abelian groups have a non-trivial inner automorphism group, and possibly also outer automorphisms. Group isomorphism problem Bijection – One-to-one...

families of groups often have a certain outer automorphism group, but in particular cases, they have other exceptional outer automorphisms. Among families...

order 2, the outer automorphism group has order 2, and the group 2.HS.2 appears as an involution centralizer in the Harada–Norton group. HS is one of...

matrix. Its outer automorphism group for n ≥ 3 is Z / 2 Z , {\displaystyle \mathbb {Z} /2\mathbb {Z} ,} while the outer automorphism group of SU(2) is...

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...

{GL} (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...

a group of Lie type, it is sometimes regarded as a 27th sporadic group. The Schur multiplier of the Tits group is trivial and its outer automorphism group...

class group of surfaces are related to various other groups, in particular braid groups and outer automorphism groups. The mapping class group appeared...

fundamental group of the (adjoint) complex form, compact real form, or any algebraic version of E7 is the cyclic group Z/2Z, and its outer automorphism group is...

^{0}({\mathfrak {g}})} is known as the outer automorphism group. It is known that the outer automorphism group for a simple Lie algebra g {\displaystyle...

four-group V. The full automorphism group of Q8 is isomorphic to S4, the symmetric group on four letters (see Matrix representations below), and the outer...

in the monster group. The outer automorphism group of B is trivial and the Schur multiplier of B has order 2. The existence of this group was suggested...

cohomology of the Weyl group W with coefficients in the maximal torus T used to define it, is related to the outer automorphism group of the normalizer N...

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...

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...