Inner Automorphism

In abstract algebra an inner automorphism is a function which, informally, involves a certain operation being applied, then another operation (shown as x below) being performed, and then the initial operation being reversed. Sometimes the initial action and its subsequent reversal change the overall result ("raise umbrella, walk through rain, lower umbrella" has a different result from just "walk through rain"), and sometimes they do not ("take off left glove, take off right glove, put on left glove" has the same effect as "take off right glove only").

More formally an inner automorphism of a group G is a function:

ƒ: G → G

defined for all x in G by

ƒ(x) = a−1xa,

where a is a given fixed element of G, and where we deem the action of group elements to occur on the right (so this would read "a times x times a^-1").

The operation a−1xa is called conjugation (see also conjugacy class).

In fact

a−1xa = 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.