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Chemistry LibreTexts

Automorphism

An isomorphism from a group (G,*) to itself is called an automorphism of this group. It is a bijection f : G → such that

f (g) * f (h) = f (g * h)

An automorphism preserves the structural properties of a group, e.g.:

  • The identity element of G is mapped to itself.
  • Subgroups are mapped to subgroups, normal subgroups to normal subgroups.
  • Conjugacy classes are mapped to conjugacy classes (the same or another).
  • The image f(g) of an element g has the same order as g.

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

Inner automorphism 

An inner automorphism of a group G is a function

fa : G → G

defined by

fa(g) = aga−1

for all g in G, where a is a given fixed element of G.

The operation aga−1 is called conjugation by a (see also conjugacy class).

The inner automorphisms form a normal subgroup of Aut(G), called the inner automorphism group and denoted by Inn(G).

The inner automorphism group is isomorphic to the quotient of G by its center Z(G). In particular, for Abelian groups the inner automorphism group consists just of the trivial automorphism.

Outer automorphism

The outer automorphism group of a group G is the quotient of the automorphism group Aut(G) by its inner automorphism group Inn(G). The outer automorphism group is usually denoted by Out(G).

For Abelian groups the mapping g → g-1 is an outer automorphism, whereas for non-Abelian groups this mapping is not even a homomorphism.