Outer automorphism group
|
|
The outer automorphism group of a group G is the quotient of the automorphism group Aut(G) by its inner automorphism group Inn(G). The outer automorphism group is usually denoted Out(G). If Out(G) is trivial, then G is said to be complete.
Note that the elements of Out(G) are not automorphisms. This is a consequence of the fact that quotients of groups are not in general subgroups. However, the elements of Aut(G) which are not inner automorphisms are usually called outer automorphisms; they map to non-trivial elements of Out(G) by the quotient map.
It was conjectured by Schreier that Out(G) is always a solvable group when G is a finite simple group. This result is now known to be true as a corollary of the classification of finite simple groups, although no simpler proof is known.
Out(G) for some finite groups
| Group | Parameter | Out(G) |
|---|---|---|
| Sn | n not equal to 6 | trivial |
| S6 | C2 | |
| An | n not equal to 6 | C2 |
| A6 | C2 x C2 | |
| Cn | n > 2 | Cnx |
| Cpn | p prime, n > 1 | GLn(p) |
| Mn | n = 11, 23, 24 | trivial |
| Mn | n = 12, 22 | C2 |
| PSL2(p) | p > 3 prime | C2 |
| PSL2(2n) | n > 1 | Cn |
| PSL3(4) = M21 | D12 | |
| Con | n = 1, 2, 3 | trivial |