# 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

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