Group Isomorphism - Automorphisms

Automorphisms

An isomorphism from a group (G,*) to itself is called an automorphism of this group. Thus it is a bijection such that

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An automorphism always maps the identity to itself. The image under an automorphism of a conjugacy class is always a conjugacy class (the same or another). The image of an element has the same order as that element.

The composition of two automorphisms is again an automorphism, and with this operation the set of all automorphisms of a group G, denoted by Aut(G), forms itself a group, the automorphism group of G.

For all Abelian groups there is at least the automorphism that replaces the group elements by their inverses. However, in groups where all elements are equal to their inverse this is the trivial automorphism, e.g. in the Klein four-group. For that group all permutations of the three non-identity elements are automorphisms, so the automorphism group is isomorphic to S₃ and Dih₃.

In Z_p for a prime number p, one non-identity element can be replaced by any other, with corresponding changes in the other elements. The automorphism group is isomorphic to Z_{p − 1}. For example, for n = 7, multiplying all elements of Z₇ by 3, modulo 7, is an automorphism of order 6 in the automorphism group, because 36 = 1 ( modulo 7 ), while lower powers do not give 1. Thus this automorphism generates Z₆. There is one more automorphism with this property: multiplying all elements of Z₇ by 5, modulo 7. Therefore, these two correspond to the elements 1 and 5 of Z₆, in that order or conversely.

The automorphism group of Z₆ is isomorphic to Z₂, because only each of the two elements 1 and 5 generate Z₆, so apart from the identity we can only interchange these.

The automorphism group of Z₂ × Z₂ × Z₂ = Dih₂ × Z₂ has order 168, as can be found as follows. All 7 non-identity elements play the same role, so we can choose which plays the role of (1,0,0). Any of the remaining 6 can be chosen to play the role of (0,1,0). This determines which corresponds to (1,1,0). For (0,0,1) we can choose from 4, which determines the rest. Thus we have 7 × 6 × 4 = 168 automorphisms. They correspond to those of the Fano plane, of which the 7 points correspond to the 7 non-identity elements. The lines connecting three points correspond to the group operation: a, b, and c on one line means a+b=c, a+c=b, and b+c=a. See also general linear group over finite fields.

For Abelian groups all automorphisms except the trivial one are called outer automorphisms.

Non-Abelian groups have a non-trivial inner automorphism group, and possibly also outer automorphisms.