Outer Automorphism

An inner automorphism of a group G is an automorphism of the form phi(g)=h^(-1)gh, where h is a fixed element of G. An outer automorphism of G is an automorphism which cannot be expressed in this form for h in G, but can be so expressed if h belongs to a larger group containing G.

For example, the automorphism of the symmetric group S_3 which maps the permutation (123) to (132) is an inner automorphism, since (132)=(12)(123)(12). However, it is an outer automorphism of the alternating group A_3 since (12) does not belong to A_3 and there is no element h of A_3 such that (132)=h^(-1)(123)h.

See also

Automorphism, Inner Automorphism

This entry contributed by David Terr

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Terr, David. "Outer Automorphism." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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