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In mathematical group theory, the Thompson replacement theorem is a theorem about the existence of certain abelian subgroups of a ''p''-group. The Glauberman replacement theorem is a generalization of it introduced by . ==Statement== Suppose that ''P'' is a finite ''p''-group for some prime ''p'', and let A be the set of abelian subgroups of ''P'' of maximal order. Suppose that ''B'' is some abelian subgroup of ''P''. The Thompson replacement theorem says that if ''A'' is an element of A that normalizes ''B'' but is not normalized by ''B'', then there is another element ''A'' * of A such that ''A'' *∩''B'' is strictly larger than ''A''∩''B'', and () normalizes ''A''. The Glauberman replacement theorem is similar, except ''p'' is assumed to be odd and the condition that ''B'' is abelian is weakened to the condition that () commutes with ''B'' and with all elements of A. Glauberman says in his paper that he does not know whether the condition that ''p'' is odd is necessary. 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「Replacement theorem」の詳細全文を読む スポンサード リンク
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