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
.