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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 Difference set ： ウィキペディア英語版 Difference set :''For the set of elements in one set but not another, see relative complement. For the set of differences of pairs of elements, see Minkowski difference.'' In combinatorics, a $(v,k,\backslash lambda)$ difference set is a subset $D$ of size $k$ of a group $G$ of order $v$ such that every nonidentity element of $G$ can be expressed as a product $d\_1d\_2^$ of elements of $D$ in exactly $\backslash lambda$ ways. A difference set $D$ is said to be ''cyclic'', ''abelian'', ''non-abelian'', etc., if the group $G$ has the corresponding property. A difference set with $\backslash lambda\; =\; 1$ is sometimes called ''planar'' or ''simple''. If $G$ is an abelian group written in additive notation, the defining condition is that every nonzero element of $G$ can be written as a ''difference'' of elements of $D$ in exactly $\backslash lambda$ ways. The term "difference set" arises in this way. ==Basic facts== * A simple counting argument shows that there are exactly $k^2-k$ pairs of elements from $D$ that will yield nonidentity elements, so every difference set must satisfy the equation $k^2-k=(v-1)\backslash lambda$. * If $D$ is a difference set, and $g\backslash in\; G$, then $gD=\backslash$ is also a difference set, and is called a translate of $D$ ($D\; +\; g$ in additive notation). * The complement of a $(v,k,\backslash lambda)$-difference set is a $(v,v-k,v-2k+\backslash lambda)$-difference set. * The set of all translates of a difference set $D$ forms a symmetric block design, called the ''development'' of $D$ and denoted by $dev(D)$. In such a design there are $v$ ''elements'' (usually called points) and $v$ ''blocks'' (subsets). Each block of the design consists of $k$ points, each point is contained in $k$ blocks. Any two blocks have exactly $\backslash lambda$ elements in common and any two points are simultaneously contained in exactly $\backslash lambda$ blocks. The group $G$ acts as an automorphism group of the design. It is sharply transitive on both points and blocks.〔. The theorem only states point transitivity, but block transitivity follows from this by the second corollary on p. 330. 〕 * * In particular, if $\backslash lambda=1$, then the difference set gives rise to a projective plane. An example of a (7,3,1) difference set in the group $\backslash mathbb/7\backslash mathbb$ is the subset $\backslash$. The translates of this difference set form the Fano plane. * Since every difference set gives a symmetric design, the parameter set must satisfy the Bruck–Ryser–Chowla theorem. * Not every symmetric design gives a difference set. 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「Difference set」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.