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In mathematics, George Glauberman's ZJ theorem states that if a finite group ''G'' is ''p''-constrained and ''p''-stable and has a normal ''p''-subgroup for some odd prime ''p'', then ''O''''p''′(''G'')''Z''(''J''(''S'')) is a normal subgroup of ''G'', for any Sylow ''p''-subgroup ''S''. ==Notation and definitions== *''J''(''S'') is the Thompson subgroup of a ''p''-group ''S'': the subgroup generated by the abelian subgroups of maximal order. *''Z''(''H'') means the center of a group ''H''. *''O''''p''′ is the maximal normal subgroup of ''G'' of order coprime to ''p'', the ''p''′-core *''O''''p'' is the maximal normal ''p''-subgroup of ''G'', the ''p''-core. *''O''''p''′,''p''(''G'') is the maximal normal ''p''-nilpotent subgroup of ''G'', the ''p''′,''p''-core, part of the upper ''p''-series. *For an odd prime ''p'', a group ''G'' with ''O''''p''(''G'') ≠ 1 is said to be ''p''-stable if whenever ''P'' is a p-subgroup of ''G'' such that ''POp′''(''G'') is normal in ''G'', and () = 1, then the image of ''x'' in N''G''(''P'')/C''G''(''P'') is contained in a normal ''p''-subgroup of N''G''(''P'')/C''G''(''P''). *For an odd prime ''p'', a group ''G'' with ''O''''p''(''G'') ≠ 1 is said to be ''p''-constrained if the centralizer C''G''(''P'') is contained in ''O''''p''′,''p''(''G'') whenever ''P'' is a Sylow ''p''-subgroup of ''O''''p''′,''p''(''G''). 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「ZJ theorem」の詳細全文を読む スポンサード リンク
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