|
|
|
|
inner automorphism
|
(Definition)
|
|
|
Let  be a group. For every  , we define a mapping
called conjugation by  . It is easy to show the conjugation map is in fact, a group automorphism.
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
is a surjective group homomorphism with kernel
 , the centre subgroup. Consequently,
 is naturally isomorphic to the quotient of
 .
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.
|
Anyone with an account can edit this entry. Please help improve it!
"inner automorphism" is owned by rmilson. [ full author list (3) | owner history (1) ]
|
|
(view preamble)
| Also defines: |
conjugation, outer, outer automorphism, automorphism group |
|
|
Cross-references: left action, right action, action, definitions, quotient, isomorphic, subgroup, centre, kernel, group homomorphism, surjective, normal subgroup, structure, operation, composition, automorphism, group automorphism, map, mapping, group
There are 66 references to this entry.
This is version 9 of inner automorphism, born on 2002-07-04, modified 2007-07-31.
Object id is 3155, canonical name is InnerAutomorphism.
Accessed 17437 times total.
Classification:
| AMS MSC: | 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
