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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]
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.
Nielsen, and later Bernhard Neumann used these ideas to give finite presentations of the automorphism groups of free groups. This is also described in (Magnus, Karrass & Solitar 2004, p. 131, Th 3.2). The automorphism group of the free group with ordered basis [ x 1, …, x n] is generated by the following 4 elementary Nielsen transformations:
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).
The automorphism group of a free abelian group of finite rank is the general linear group (,), which can be described concretely (for a specific basis of the free automorphism group) as the set of invertible integer matrices under the operation of matrix multiplication.
Download as PDF; Printable version; In other projects ... ATLAS of Finite Groups; Automorphisms of the symmetric and alternating groups; B. ... Special abelian ...
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 ...
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