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In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Ludwig Sylow () that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups. For a prime number ''p'', a Sylow ''p''-subgroup (sometimes ''p''-Sylow subgroup) of a group ''G'' is a maximal ''p''-subgroup of ''G'', i.e. a subgroup of ''G'' that is a ''p''-group (so that the order of any group element is a power of ''p'') that is not a proper subgroup of any other ''p''-subgroup of ''G''. The set of all Sylow ''p''-subgroups for a given prime ''p'' is sometimes written Syl''p''(''G''). The Sylow theorems assert a partial converse to Lagrange's theorem. While Lagrange's theorem states that for any finite group ''G'' the order (number of elements) of every subgroup of ''G'' divides the order of ''G'', the Sylow theorems state that for any prime factor ''p'' of the order of a finite group ''G'', there exists a Sylow ''p''-subgroup of ''G''. The order of a Sylow ''p''-subgroup of a finite group ''G'' is ''pn'', where ''n'' is the multiplicity of ''p'' in the order of ''G'', and any subgroup of order ''pn'' is a Sylow ''p''-subgroup of ''G''. The Sylow ''p''-subgroups of a group (for a given prime ''p'') are conjugate to each other. The number of Sylow ''p''-subgroups of a group for a given prime ''p'' is congruent to == Theorems == Collections of subgroups that are each maximal in one sense or another are common in group theory. The surprising result here is that in the case of Syl''p''(''G''), all members are actually isomorphic to each other and have the largest possible order: if |''G''| = ''pnm'' with ''n'' > 0 where ''p'' does not divide ''m'', then any Sylow ''p''-subgroup ''P'' has order |''P''| = ''pn''. That is, ''P'' is a ''p''-group and gcd(|''G'' : ''P''|, ''p'') = 1. These properties can be exploited to further analyze the structure of ''G''. The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in ''Mathematische Annalen''. Theorem 1: For any prime factor ''p'' with multiplicity ''n'' of the order of a finite group ''G'', there exists a Sylow ''p''-subgroup of ''G'', of order ''pn''. The following weaker version of theorem 1 was first proved by Cauchy, and is known as Cauchy's theorem. Corollary: Given a finite group ''G'' and a prime number ''p'' dividing the order of ''G'', then there exists an element (and hence a subgroup) of order ''p'' in ''G''.〔Fraleigh, Victor J. Katz. A First Course In Abstract Algebra. p. 322. ISBN 9788178089973〕 Theorem 2: Given a finite group ''G'' and a prime number ''p'', all Sylow ''p''-subgroups of ''G'' are conjugate to each other, i.e. if ''H'' and ''K'' are Sylow ''p''-subgroups of ''G'', then there exists an element ''g'' in ''G'' with ''g''−1''Hg'' = ''K''. Theorem 3: Let ''p'' be a prime factor with multiplicity ''n'' of the order of a finite group ''G'', so that the order of ''G'' can be written as , where and ''p'' does not divide ''m''. Let ''np'' be the number of Sylow ''p''-subgroups of ''G''. Then the following hold: * ''np'' divides ''m'', which is the index of the Sylow ''p''-subgroup in ''G''. * ''np'' ≡ 1 mod ''p''. * ''np'' = |''G'' : ''NG''(''P'')|, where ''P'' is any Sylow ''p''-subgroup of ''G'' and ''NG'' denotes the normalizer. 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「Sylow theorems」の詳細全文を読む スポンサード リンク
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