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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