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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
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