Inner automorphism
An inner automorphism is an automorphism on a group of the form
, for some
in
. This mapping is denoted
. Every such mapping is an automorphism.
Sometimes is denoted as
, or as
.
Theorem. For every in
,
is a group automorphism on
. Furthermore, the mapping
is a group homomorphism from
to
, the group of automorphisms on
. Its kernel is the center of
, and its image, the set of inner automorphisms, is a normal subgroup of
.
Proof. Let be an element of
. Since
is a group,
if and only if
, so
is injective. Every element
has an inverse image
as well, so
is surjective onto
. Finally, for all
,
so
is an endomorphism of
. Therefore it is an automorphism of
.
Since
is a group homomorphism from
to
. Note that
is the identity map on
if and only if, for all
,
, which is true if and only if
, which is true if and only if
is in the center of
.
Finally, if is any automorphism of
,
is an inner automorphism on
, and
is any element of
, then
Thus
is a normal subgroup of
.
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