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In group theory, a branch of mathematics, Frattini's argument is an important lemma in the structure theory of finite groups. It is named after Giovanni Frattini, who first used it in a paper from 1885 when defining the Frattini subgroup of a group. ==Statement and proof== Frattini's Argument. If ''G'' is a finite group with normal subgroup ''H'', and if ''P'' is a Sylow ''p''-subgroup of ''H'', then Proof: ''P'' is a Sylow ''p''-subgroup of ''H'', so every Sylow ''p''-subgroup of ''H'' is an ''H''-conjugate ''h''−1''Ph'' for some ''h'' ∈ ''H'' (see Sylow theorems). Let ''g'' be any element of ''G''. Since ''H'' is normal in ''G'', the subgroup ''g''−1''Pg'' is contained in ''H''. This means that ''g''−1''Pg'' is a Sylow ''p''-subgroup of ''H''. Then by the above, it must be ''H''-conjugate to ''P'': that is, for some ''h'' ∈ ''H'' :''g''−1''Pg'' = ''h''−1''Ph'', so :''hg''−1''Pgh''−1 = ''P''; thus :''gh''−1 ∈ ''N''''G''(''P''), and therefore ''g'' ∈ ''N''''G''(''P'')''H''. But ''g'' ∈ ''G'' was arbitrary, so ''G'' = ''HN''''G''(''P'') = ''N''''G''(''P'')''H''. 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「Frattini's argument」の詳細全文を読む スポンサード リンク
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