Inner automorphism

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

defined by

where a is a given fixed element of G.

The operation a−1xa 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 conjugation has a net effect ("take off shoes, take off socks, replace shoes"), and sometimes it does not ("take off left glove, take off right glove, replace left glove" or "take off right glove" are equivalent).

In fact

is equivalent to saying

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.

Contents

Notation

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

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 (xy)a = xaya.

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

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).

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