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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Herzog–Schönheim conjecture ： ウィキペディア英語版 Herzog–Schönheim conjecture In mathematics, the Herzog–Schönheim conjecture is a combinatorial problem in the area of group theory, posed by Marcel Herzog and Jochanan Schönheim in 1974.〔. As cited by .〕 Let $G$ be a group, and let :$A=\backslash$ be a finite system of left cosets of subgroups $G\_1,\backslash ldots,G\_k$ of $G$. Herzog and Schönheim conjectured that if $A$ forms a partition of $G$ with $k>1$, then the (finite) indices $(),\backslash ldots,()$ cannot be distinct. In contrast, if repeated indices are allowed, then partitioning a group into cosets is easy: if $H$ is any subgroup of $G$ with index then $G$ can be partitioned into $k$ left cosets of $H$. ==Subnormal subgroups== In 2004 Zhi-Wei Sun proved an extended version of the Herzog–Schönheim conjecture in the case where $G\_1,\backslash ldots,G\_k$ are subnormal in $G$.〔.〕 A basic lemma in Sun's proof states that if $G\_1,\backslash ldots,G\_k$ are subnormal and of finite index in $G$, then :$\backslash bigg()\backslash \; \backslash bigg|\backslash \; \backslash prod\_^k()$ and hence :$P\backslash bigg(\backslash bigg()\backslash \; \backslash bigg)$ =\bigcup_^kP(()), where $P(n)$ denotes the set of prime divisors of $n$. 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「Herzog–Schönheim conjecture」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.