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Groupprops β

Outer automorphism group maps to automorphism group of center

Statement

Suppose G is a group. Denote by \operatorname{Out}(G) the outer automorphism group of G and by Z(G) the center of G. Denote by \operatorname{Aut}(Z(G)) the automorphism group of Z(G). Then, there is a canonical homomorphism:

\operatorname{Out}(G) \to \operatorname{Aut}(Z(G))

defined as follows: for any outer automorphism class [\sigma] of G, pick a representative automorphism \sigma, and consider the restriction of \sigma to Z(G).

Proof that this is well defined: Since any inner automorphism restricts to the identity on Z(G), the automorphism obtained by restriction is independent of the choice of representative.

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