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The fundamental theorem of finite abelian groups states that every finite abelian group can be expressed as the direct sum of cyclic subgroups of prime-power order; it is also known as the basis theorem for finite abelian groups. Moreover, automorphism groups of cyclic groups are examples of abelian groups. [13]
The outer automorphism group of a finite simple group of Lie type is an extension of a group of "diagonal automorphisms" (cyclic except for D n (q), when it has order 4), a group of "field automorphisms" (always cyclic), and a group of "graph automorphisms" (of order 1 or 2 except for D 4 (q), when it is the symmetric group on 3 points).
In mathematics, the automorphism group of an object X is the group consisting of automorphisms of X under composition of morphisms.For example, if X is a finite-dimensional vector space, then the automorphism group of X is the group of invertible linear transformations from X to itself (the general linear group of X).
A result of Wolfgang Gaschütz says that if G is a finite non-abelian p-group, then G has an automorphism of p-power order which is not inner. It is an open problem whether every non-abelian p-group G has an automorphism of order p. The latter question has positive answer whenever G has one of the following conditions: G is nilpotent of class 2
This also yields another outer automorphism of A 6, and this is the only exceptional outer automorphism of a finite simple group: [4] for the infinite families of simple groups, there are formulas for the number of outer automorphisms, and the simple group of order 360, thought of as A 6, would be expected to have two outer automorphisms, not four.
The study of automorphisms of algebraic field extensions is the starting point and the main object of Galois theory. The automorphism group of the quaternions (H) as a ring are the inner automorphisms, by the Skolem–Noether theorem: maps of the form a ↦ bab −1. [4] This group is isomorphic to SO(3), the group of rotations in 3-dimensional ...
The fundamental theorem of finitely generated abelian groups can be stated two ways, generalizing the two forms of the fundamental theorem of finite abelian groups.The theorem, in both forms, in turn generalizes to the structure theorem for finitely generated modules over a principal ideal domain, which in turn admits further generalizations.
This in fact characterizes elementary abelian groups among all finite groups: if G is a finite group with identity e such that Aut(G) acts transitively on G \ {e}, then G is elementary abelian. (Proof: if Aut(G) acts transitively on G \ {e}, then all nonidentity elements of G have the same (necessarily prime) order.