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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 Descendant tree (group theory) ： ウィキペディア英語版 Descendant tree (group theory) In mathematics, specifically group theory, a descendant tree is a hierarchical structure for visualizing parent-descendant relations between isomorphism classes of finite groups of prime power order $p^n$, for a fixed prime number $p$ and varying integer exponents $n\backslash ge\; 0$. Such groups are briefly called ''finite'' ''p-groups''. The ''vertices'' of a descendant tree are isomorphism classes of finite ''p''-groups. Additionally to their ''order'' $p^n$, finite ''p''-groups have two further related invariants, the ''nilpotency class'' $c$ and the coclass $r=n-c$. It turned out that descendant trees of a particular kind, the so-called pruned coclass trees whose infinitely many vertices share a common coclass $r$, reveal a repeating finite pattern. These two crucial properties of finiteness and periodicity admit a characterization of all members of the tree by finitely many parametrized presentations. Consequently, descendant trees play a fundamental role in the classification of finite ''p''-groups. By means of kernels and targets of Artin transfer homomorphisms, descendant trees can be endowed with additional structure. An important question is how the descendant tree $\backslash mathcal(R)$ can actually be constructed for an assigned starting group which is taken as the root $R$ of the tree. The ''p''-group generation algorithm is a recursive process for constructing the descendant tree of a forgiven finite ''p''-group playing the role of the tree root. This algorithm is implemented in the computational algebra systems GAP and Magma. ==Definitions and terminology== According to M. F. Newman ,〔 〕 there exist several distinct definitions of the parent $\backslash pi(G)$ of a finite ''p''-group $G$. The common principle is to form the quotient $\backslash pi(G)=G/N$ of $G$ by a suitable normal subgroup $N\backslash triangleleft\; G$ which can be either :# the centre $N=\backslash zeta\_1(G)$ of $G$, whence $\backslash pi(G)=G/\backslash zeta\_1(G)$ is called the ''central quotient'' of $G$, or :# the last non-trivial term $N=\backslash gamma\_c(G)$ of the lower central series of $G$, where $c$ denotes the nilpotency class of $G$, or :# the last non-trivial term $N=P\_(G)$ of the lower exponent-''p'' central series of $G$, where $c$ denotes the exponent-''p'' class of $G$, or :# the last non-trivial term $N=G^$ of the derived series of $G$, where $d$ denotes the derived length of $G$. In each case, $G$ is called an immediate descendant of $\backslash pi(G)$ and a ''directed edge'' of the tree is defined either by $G\backslash to\backslash pi(G)$ in the direction of the canonical projection $\backslash pi:G\backslash to\backslash pi(G)$ onto the quotient $\backslash pi(G)=G/N$ or by $\backslash pi(G)\backslash to\; G$ in the opposite direction, which is more usual for descendant trees. The former convention is adopted by C. R. Leedham-Green and M. F. Newman ,〔 〕 by M. du Sautoy and D. Segal ,〔 〕 by C. R. Leedham-Green and S. McKay ,〔 〕 and by B. Eick, C. R. Leedham-Green, M. F. Newman and E. A. O'Brien .〔 〕 The latter definition is used by M. F. Newman ,〔 by M. F. Newman and E. A. O'Brien ,〔 〕 by M. du Sautoy ,〔 〕 and by B. Eick and C. R. Leedham-Green .〔 〕 In the following, the direction of the canonical projections is selected for all edges. Then, more generally, a vertex $R$ is a descendant of a vertex $P$, and $P$ is an ancestor of $R$, if either $R$ is equal to $P$ or there is a ''path'' $(1)\backslash qquad\; R=Q\_0\backslash to\; Q\_1\backslash to\backslash cdots\backslash to\; Q\_\backslash to\; Q\_m=P$, with $m\backslash ge\; 1$, of directed edges from $R$ to $P$. The vertices forming the path necessarily coincide with the ''iterated parents'' $Q\_j=\backslash pi^(R)$ of $R$, with $0\backslash le\; j\backslash le\; m$: $(2)\backslash qquad\; R=\backslash pi^(R)\backslash to\backslash pi^(R)\backslash to\backslash cdots\backslash to\backslash pi^(R)\backslash to\backslash pi^(R)=P$, with $m\backslash ge\; 1$, In the most important special case (P2) of parents defined as last non-trivial lower central quotients, they can also be viewed as the successive ''quotients'' $R/\backslash gamma\_(R)$ ''of class'' $c-j$ of $R$ when the nilpotency class of $R$ is given by $c\backslash ge\; m$: $(3)\backslash qquad\; R\backslash simeq\; R/\backslash gamma\_(R)\backslash to\; R/\backslash gamma\_(R)\backslash to\backslash cdots\backslash to\; R/\backslash gamma\_(R)\backslash to\; R/\backslash gamma\_(R)\backslash simeq\; P$, with $c\backslash ge\; m\backslash ge\; 1$. Generally, the descendant tree $\backslash mathcal(G)$ of a vertex $G$ is the subtree of all descendants of $G$, starting at the root $G$. The maximal possible descendant tree $\backslash mathcal(1)$ of the trivial group $1$ contains all finite ''p''-groups and is somewhat exceptional, since, for any parent definition (P1–P4), the trivial group $1$ has infinitely many abelian ''p''-groups as its immediate descendants. The parent definitions (P2–P3) have the advantage that any non-trivial finite ''p''-group (of order divisible by $p$) possesses only finitely many immediate descendants. 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「Descendant tree (group theory)」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.