Let be a group. For every , we define a mapping
An automorphism of that corresponds to conjugation by some is called inner. An automorphism that isn’t inner is called an outer automorphism.
The composition operation gives the set of all automorphisms of the structure of a group, . The inner automorphisms also form a group, , which is a normal subgroup of . Indeed, if is an inner automorphism and an arbitrary automorphism, then
Let us also note that the mapping
Note: the above definitions and assertions hold, mutatis mutandi, if we define the conjugation action of on to be the right action
rather than the left action given above.
|Date of creation||2013-03-22 12:49:53|
|Last modified on||2013-03-22 12:49:53|
|Last modified by||rmilson (146)|