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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 P-group generation algorithm ： ウィキペディア英語版 P-group generation algorithm In mathematics, specifically group theory, finite groups of prime power order $p^n$, for a fixed prime number $p$ and varying integer exponents $n\backslash ge\; 0$, are briefly called ''finite'' ''p-groups''. The ''p''-group generation algorithm by M. F. Newman 〔 〕 and E. A. O'Brien 〔 〕 〔 〕 is a recursive process for constructing the descendant tree of an assigned finite ''p''-group which is taken as the root of the tree. ==Lower exponent-''p'' central series== For a finite ''p''-group $G$, the lower exponent-''p'' central series (briefly lower ''p''-central series) of $G$ is a descending series $(P\_j(G))\_$ of characteristic subgroups of $G$, defined recursively by $(1)\backslash qquad\; P\_0(G):=G$ and $P\_j(G):=\backslash lbrack\; P\_(G),G\backslash rbrack\backslash cdot\; P\_(G)^p$, for $j\backslash ge\; 1$. Since any non-trivial finite ''p''-group $G>1$ is nilpotent, there exists an integer $c\backslash ge\; 1$ such that $P\_(G)>P\_c(G)=1$ and $\backslash mathrm\_p(G):=c$ is called the exponent-''p'' class (briefly ''p''-class) of $G$. Only the trivial group $1$ has $\backslash mathrm\_p(1)=0$. Generally, for any finite ''p''-group $G$, its ''p''-class can be defined as $\backslash mathrm\_p(G):=\backslash min\backslash lbrace\; c\backslash ge\; 0\backslash mid\; P\_c(G)=1\backslash rbrace$. The complete lower ''p''-central series of $G$ is therefore given by $(2)\backslash qquad\; G=P\_0(G)>\backslash Phi(G)=P\_1(G)>P\_2(G)>\backslash cdots>P\_(G)>P\_c(G)=1$, since $P\_1(G)=\backslash lbrack\; P\_0(G),G\backslash rbrack\backslash cdot\; P\_0(G)^p=\backslash lbrack\; G,G\backslash rbrack\backslash cdot\; G^p=\backslash Phi(G)$ is the Frattini subgroup of $G$. For the convenience of the reader and for pointing out the shifted numeration, we recall that the (usual) lower central series of $G$ is also a descending series $(\backslash gamma\_j(G))\_$ of characteristic subgroups of $G$, defined recursively by $(3)\backslash qquad\; \backslash gamma\_1(G):=G$ and $\backslash gamma\_j(G):=\backslash lbrack\backslash gamma\_(G),G\backslash rbrack$, for $j\backslash ge\; 2$. As above, for any non-trivial finite ''p''-group $G>1$, there exists an integer $c\backslash ge\; 1$ such that $\backslash gamma\_c(G)>\backslash gamma\_(G)=1$ and $\backslash mathrm(G):=c$ is called the nilpotency class of $G$, whereas $c+1$ is called the index of nilpotency of $G$. Only the trivial group $1$ has $\backslash mathrm(1)=0$. The complete lower central series of $G$ is given by $(4)\backslash qquad\; G=\backslash gamma\_1(G)>G^=\backslash gamma\_2(G)>\backslash gamma\_3(G)>\backslash cdots>\backslash gamma\_c(G)>\backslash gamma\_(G)=1$, since $\backslash gamma\_2(G)=\backslash lbrack\backslash gamma\_1(G),G\backslash rbrack=\backslash lbrack\; G,G\backslash rbrack=G^$ is the commutator subgroup or derived subgroup of $G$. The following Rules should be remembered for the exponent-''p'' class: Let $G$ be a finite ''p''-group. :# Rule: $\backslash mathrm(G)\backslash le\backslash mathrm\_p(G)$, since the $\backslash gamma\_j(G)$ descend more quickly than the $P\_j(G)$. :# Rule: If $\backslash vartheta\backslash in\backslash mathrm(G,\backslash tilde)$, for some group $\backslash tilde$, then $\backslash vartheta(P\_j(G))=P\_j(\backslash vartheta(G))$, for any $j\backslash ge\; 0$. :# Rule: For any $c\backslash ge\; 0$, the conditions $N\backslash triangleleft\; G$ and $\backslash mathrm\_p(G/N)=c$ imply $P\_c(G)\backslash le\; N$. :# Rule: Let $c\backslash ge\; 0$. If $\backslash mathrm\_p(G)=c$, then $\backslash mathrm\_p(G/P\_k(G))=\backslash min(k,c)$, for all $k\backslash ge\; 0$, in particular, $\backslash mathrm\_p(G/P\_k(G))=k$, for all $0\backslash le\; k\backslash le\; c$. 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「P-group generation algorithm」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.