Inner automorphism

In abstract algebra, an inner automorphism of a group is a function f : G -> G defined by

f(x) = axa^-1    

for all x in G. As the name suggests, f is an automorphism of G.

The collection of all inner automorphisms of G forms a normal subgroup of the full automorphism group G. This group is denoted by Inn(G).

By associating the element a in G with the inner automorphism f in Inn(G) as above, one obtains an isomorphism between the factor group G/Z(G) (where Z(G) is the center of G) and Inn(G). As a consequence, the group of inner automorphisms Inn(G) is trivial (i.e. consists only of the identity element) if and only if G is abelian.