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In group theory, the conjugate closure of a subset ''S'' of a group ''G'' is the subgroup of ''G'' generated by ''S''''G'', i.e. the closure of ''S''''G'' under the group operation, where ''S''''G'' is the set of the conjugates of the elements of ''S'': :''S''''G'' = The conjugate closure of ''S'' is denoted <''S''''G''> or <''S''>''G''. The conjugate closure of any subset ''S'' of a group ''G'' is always a normal subgroup of ''G''; in fact, it is the smallest (by inclusion) normal subgroup of ''G'' which contains ''S''. For this reason, the conjugate closure is also called the normal closure of ''S'' or the normal subgroup generated by ''S''. The normal closure can also be characterized as the intersection of all normal subgroups of ''G'' which contain ''S''. Any normal subgroup is equal to its normal closure. The conjugate closure of a singleton subset of a group ''G'' is a normal subgroup generated by ''a'' and all elements of ''G'' which are conjugate to ''a''. Therefore, any simple group is the conjugate closure of any non-identity group element. The conjugate closure of the empty set is the trivial group. Contrast the normal closure of ''S'' with the ''normalizer'' of ''S'', which is (for ''S'' a group) the largest subgroup of ''G'' in which ''S'' ''itself'' is normal. (This need not be normal in the larger group ''G'', just as <''S''> need not be normal in its conjugate/normal closure.) Dual to the concept of normal closure is that of ''normal interior'' or ''normal core'', defined as the join of all normal subgroups contained in ''S''.〔Robinson p.16〕 ==References== * * 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「Conjugate closure」の詳細全文を読む スポンサード リンク
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