Conjugation map

In abstract algebra an inner automorphism is a certain type of automorphism of a group defined in terms of a fixed element of the group, called the conjugating element. Formally, if G is a group and a is an element of G, then the inner automorphism defined by a is the map f from G to itself defined for all x in G by the formula

Here we use the convention that group elements act on the right.

The operation x ↦ a⁻¹xa is called conjugation (see also conjugacy class), and it is often of interest to distinguish the cases where conjugation by one element leaves another element unchanged from cases where conjugation generates a new element.

In fact, saying that conjugation of x by a leaves x unchanged is equivalent to saying that a and x commute:

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.

An automorphism of a group G is inner if and only if it extends to every group containing G.

The expression a⁻¹xa is often denoted exponentially by x^a. This notation is used because we have the rule (x^a)^b = x^ab (giving a right action of G on itself).

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 that (xy)^a = x^ay^a.

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