Editing Group isomorphism (section)

== Automorphisms ==<!-- This section is linked from [[Abelian group]] -->

An isomorphism from a group <math>(G, *)</math> to itself is called an [[automorphism]] of the group. Thus it is a bijection <math>f : G \to G</math> such that
<math display="block">f(u) * f(v) = f(u * v).</math>

The [[image (mathematics)|image]] under an automorphism of a [[conjugacy class]] is always a conjugacy class (the same or another).

The [[function composition|composition]] of two automorphisms is again an automorphism, and with this operation the set of all automorphisms of a group <math>G,</math> denoted by <math>\operatorname{Aut}(G),</math> itself forms a group, the ''[[automorphism group]]'' of <math>G.</math>

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 inverses this is the [[trivial automorphism]], e.g. in the [[Klein four-group]]. For that group all [[permutation]]s of the three non-identity elements are automorphisms, so the automorphism group is isomorphic to <math>S_3</math> (which itself is isomorphic to <math>\operatorname{Dih}_3</math>).

In <math>\Z_p</math> for a [[prime number]] <math>p,</math> one non-identity element can be replaced by any other, with corresponding changes in the other elements. The automorphism group is isomorphic to <math>\Z_{p-1}</math> For example, for <math>n = 7,</math> multiplying all elements of <math>\Z_7</math> by 3, modulo 7, is an automorphism of order 6 in the automorphism group, because <math>3^6 \equiv 1 \pmod 7,</math> while lower powers do not give 1. Thus this automorphism generates <math>\Z_6.</math> There is one more automorphism with this property: multiplying all elements of <math>\Z_7</math> by 5, modulo 7. Therefore, these two correspond to the elements 1 and 5 of <math>\Z_6,</math> in that order or conversely.

The automorphism group of <math>\Z_6</math> is isomorphic to <math>\Z_2,</math> because only each of the two elements 1 and 5 generate <math>\Z_6,</math> so apart from the identity we can only interchange these.

The automorphism group of <math>\Z_2 \oplus \Z_2 \oplus \Z_2 = \operatorname{Dih}_2 \oplus \Z_2</math> 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 <math>(1,0,0).</math> Any of the remaining 6 can be chosen to play the role of (0,1,0). This determines which element corresponds to <math>(1,1,0).</math> For <math>(0,0,1)</math> we can choose from 4, which determines the rest. Thus we have <math>7 \times 6 \times 4 = 168</math> automorphisms. They correspond to those of the [[Fano plane]], of which the 7 points correspond to the 7 {{nowrap|non-identity}} elements. The lines connecting three points correspond to the group operation: <math>a, b,</math> and <math>c</math> on one line means <math>a + b = c,</math> <math>a + c = b,</math> and <math>b + c = a.</math> See also [[General linear group#Over finite fields|general linear group over finite fields]].

For abelian groups, all non-trivial automorphisms are [[outer automorphism]]s.

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