outer automorphism group
The outer automorphism group of a group is the quotient
(http://planetmath.org/QuotientGroup) of its automorphism group
by its inner automorphism group:
There is some variance in terminology about “an outer automorphism.” Some authors define an outer automorphism as any automorphism which is not an inner automorphism. In this way an outer automorphism still permutes the group . However, an equally common definition is to declare an outer automorphism as an element of and consequently the elements are cosets of , and not a map . In this definition it is not generally possible to treat the element as a permutation
of . In particular, the outer automorphism group of a general group does not act on the group in a natural way. An exception is when is abelian
so that ; thus, the elements of are canonically identified with those of so we can speak of the action by outer automorphisms.