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^{−1}xa 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^{−1}xa 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).
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} = x^{a}y^{a}.
Inner and outer automorphism groups
The composition of two inner automorphisms is again an inner automorphism (as mentioned above: (x^{a})^{b}=x^{ab}, 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 noninner automorphism yields a nontrivial element of Out(G), but different noninner automorphisms may yield the same element of Out(G).
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