Inner automorphism

From Free net encyclopedia

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.

The operation axa^-1 is called conjugation (see also conjugacy class).

It is often denoted exponentially by ^ax. This notation is used because we have the rule ^a(^bx)=^abx (giving a left action of G on itself). An alternative form, leading to a right action, can be obtained by (re)defining f(x) to be a^-1xa; this form is denoted x^a

As the name suggests, f is an automorphism of G. An automorphism not of this form is called an outer automorphism.

The collection of all inner automorphisms of G is a group, denoted Inn(G). It 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 elements of that group are cosets of automorphisms, and hence are not actually the outer automorphisms, since those can't form a group.)

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 Inn(G) of inner automorphisms is trivial (i.e. consists only of the identity element) if and only if G is abelian. Inn(G) can only be a cyclic group when it is trivial.

At the opposite end of the spectrum, it is possible that the inner automorphisms exhaust the entire automorphism group---a group whose automorphisms are all inner is called complete.

An automorphism of a Lie algebra <math>\mathfrak{g}</math> is called an inner automorphism if it is of the form Ad_g, where Ad is the adjoint map and g is an element of a Lie group whose Lie algebra is <math>\mathfrak{g}</math>. The notion of inner automorphism for Lie algebras is compatible with the notion for groups in the sense that an inner automorphism of a Lie group induces a unique inner automorphism of the corresponding Lie algebra.