Inner Automorphism - Inner and Outer Automorphism Groups

Inner and Outer Automorphism Groups

The composition of two inner automorphisms is again an inner automorphism (as mentioned above: (xa)b=xab, and with this operation, the collection of all inner automorphisms of G is itself a group, the inner automorphism group of G denoted Inn(G).

Inn(G) is a normal subgroup of the full automorphism group Aut(G) of G. The quotient group

Aut(G)/Inn(G)

is known as the outer automorphism group Out(G). The outer automorphism group measures, in a sense, how many automorphisms of G are not inner. Every non-inner automorphism yields a non-trivial element of Out(G), but different non-inner automorphisms may yield the same element of Out(G).

By associating the element a in G with the inner automorphism ƒ(x) = xa in Inn(G) as above, one obtains an isomorphism between the quotient group G/Z(G) (where Z(G) is the center of G) and the inner automorphism group:

G/Z(G) = Inn(G).

This is a consequence of the first isomorphism theorem, because Z(G) is precisely the set of those elements of G that give the identity mapping as corresponding inner automorphism (conjugation changes nothing).