# Outer automorphism group maps to automorphism group of center

From Groupprops

## Statement

Suppose is a group. Denote by the outer automorphism group of and by the center of . Denote by the automorphism group of . Then, there is a canonical homomorphism:

defined as follows: for any outer automorphism class of , pick a representative automorphism , and consider the restriction of to .

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