Quotient group maps to outer automorphism group of normal subgroup

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Statement

Suppose G is a group and N is a normal subgroup of G. There is a natural choice of homomorphism of groups from the quotient group G/N to the outer automorphism group \operatorname{Out}(N):

G/N \to \operatorname{Out}(N)

defined as follows: for any element of G/N, pick an element g \in G in that coset of N. Conjugation by g induces an automorphism of N, i.e., an element of the automorphism group \operatorname{Aut}(N). Although the automorphism depends on the choice of g in the coset of N, the coset of \operatorname{Inn}(N) in \operatorname{Aut}(N) for that element is independent of g.

More explicitly, there is a composite map:

G \to \operatorname{Inn}(G) \to \operatorname{Aut}(N)

Under this map, the image of the subgroup N of G lies inside \operatorname{Inn}(N). Thus, we get an induced map from the quotient group G/N to the quotient group \operatorname{Aut}(N)/\operatorname{Inn}(N) = \operatorname{Out}(N).