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In mathematical group theory, the Hall–Higman theorem, due to , describes the possibilities for the minimal polynomial of an element of prime power order for a representation of a ''p''-solvable group. ==Statement== Suppose that ''G'' is a ''p''-solvable group with no normal ''p''-subgroups, acting faithfully on a vector space over a field of characteristic ''p''. If ''x'' is an element of order ''p''''n'' of ''G'' then the minimal polynomial is of the form (''X'' − 1)''r'' for some ''r'' ≤ ''p''''n''. The Hall–Higman theorem states that one of the following 3 possibilities holds: *''r'' = ''p''''n'' *''p'' is a Fermat prime and the Sylow 2-subgroups of ''G'' are non-abelian and ''r'' ≥ ''p''''n'' −''p''''n''−1 *''p'' = 2 and the Sylow ''q''-subgroups of ''G'' are non-abelian for some Mersenne prime ''q'' = 2''m'' − 1 less than 2''n'' and ''r'' ≥ 2''n'' − 2''n''−''m''. 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「Hall–Higman theorem」の詳細全文を読む スポンサード リンク
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