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Cauchy's theorem is a theorem in the mathematics of group theory, named after Augustin Louis Cauchy. It states that if ''G'' is a finite group and ''p'' is a prime number dividing the order of ''G'' (the number of elements in ''G''), then ''G'' contains an element of order ''p''. That is, there is ''x'' in ''G'' so that ''p'' is the lowest non-zero number with ''x''''p'' = ''e'', where ''e'' is the identity element. The theorem is related to Lagrange's theorem, which states that the order of any subgroup of a finite group ''G'' divides the order of ''G''. Cauchy's theorem implies that for any prime divisor ''p'' of the order of ''G'', there is a subgroup of ''G'' whose order is ''p''—the cyclic group generated by the element in Cauchy's theorem. Cauchy's theorem is generalised by Sylow's first theorem, which implies that if ''p''''n'' is any prime power dividing the order of ''G'', then G has a subgroup of order ''p''''n''. ==Statement and proof== Many texts appear to prove the theorem with the use of strong induction and the class equation, though considerably less machinery is required to prove the theorem in the abelian case. One can also invoke group actions for the proof. Theorem: Let ''G'' be a finite group and ''p'' be a prime. If ''p'' divides the order of ''G'', then ''G'' has an element of order ''p''. 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「Cauchy's theorem (group theory)」の詳細全文を読む スポンサード リンク
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