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In abstract algebra an inner automorphism is a function in which an initial operation is applied, then another operation, and then the initial operation is reversed. With letters to indicate the operations and thing-being-transformed: . 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 only "take off right glove"). More formally an inner automorphism of a group ''G'' is a function :''f'' : ''G'' → ''G'' defined for all ''x'' in ''G'' by :''f''(''x'') = ''a''−1''xa'', where ''a'' is a given fixed element of ''G'', and where we deem the action of group elements to occur on the right. The operation ''a''−1''xa'' 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 (as in the "gloves" analogy above) from cases where conjugation generates a new element (as in the "umbrella" analogy). In fact, saying, conjugation of ''x'' by ''a'' leaves ''x'' unchanged, i.e. :''a''−1''xa'' = ''x'', is equivalent to saying that ''a'' and ''x'' commute: :''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. An automorphism of a group ''G'' is inner if and only if it extends to every group containing ''G''. ==Notation== The expression ''a''−1''xa'' is often denoted exponentially by ''xa''. This notation is used because we have the rule (giving a right action of ''G'' on itself). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Inner automorphism」の詳細全文を読む スポンサード リンク
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