# Quotient group maps to outer automorphism group of normal subgroup

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## Statement

Suppose is a group and is a normal subgroup of . There is a natural choice of homomorphism of groups from the quotient group to the outer automorphism group :

defined as follows: for any element of , pick an element in that coset of . Conjugation by induces an automorphism of , i.e., an element of the automorphism group . Although the automorphism depends on the choice of in the coset of , the coset of in for that element is independent of .

More explicitly, there is a composite map:

Under this map, the image of the subgroup of lies inside . Thus, we get an induced map from the quotient group to the quotient group .