Inner automorphism

In abstract algebra, an inner automorphism of a group G is a function

ƒ: G → G

defined by

ƒ(x) = axa⁻¹, for all x in G,

where a is a given fixed element of G.

The operation axa⁻¹ is called conjugation (see also conjugacy class). Informally, in a conjugation a certain operation is applied, then another one (x) is carried out, and then the initial operation is reversed. Sometimes this matters ('take off shoes, take off socks, replace shoes'), and sometimes ('take off left glove, take off right glove, replace left glove') it doesn't.

In fact

axa⁻¹ = x

is equivalent to saying

ax = xa.

Therefore the existence and number of inner automorphisms that are not the identity mapping is a kind of measure of the failure of the commutative law in the group. This is one good reason to study this concept in group theory.

Notation

The expression axa⁻¹ 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 denoting a⁻¹xa as x^a.

Properties

Every inner automorphism is indeed an automorphism of the group G, i.e. it is a bijective map from G to G and it is a homomorphism; meaning ^a(xy) = ^ax^ay.

The inner and outer automorphism groups

The composition of two inner automorphisms is again an inner automorphism (as mentioned above: ^a(^bx)=^abx), 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).

An automorphism of G which is not inner is called an outer automorphism.

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. Note however that the elements of Out(G) are not the outer automorphisms (which do not form a group) but are cosets of automorphisms. Every outer automorphism yields a non-trivial element of Out(G), but different outer automorphisms may yield the same element of Out(G).

By associating the element a in G with the inner automorphism ƒ(x) = ^ax 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).

Types of groups

It follows that the group Inn(G) of inner automorphisms is itself 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, by a basic result on the center of a group.

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.

If the inner automorphism group of a perfect group G is simple, then G is called quasisimple.

Ring case

Given a ring R and a unit u in R, the map ƒ(x) = uxu^-1 is a ring automorphism of R. The ring automorphisms of this form are called inner automorphisms of R. They form a normal subgroup of the automorphism group of R.

Lie algebra case

An automorphism of a Lie algebra 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 . 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.

Extension

If G arises as the group of units of a ring A, then an inner automorphism on G can be extended to a projectivity on the projective space over A by inversive ring geometry. In particular, the inner automorphisms of the classical linear groups can be so extended.