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Quelle main.tex Sprache: Latech |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Chapter{Methods and functions} This chapter describes all the main methods and functions of this package. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Functions for finite $p$-groups} Let $G$ be a finite $p$-group given by a consistent polycyclic presentation as Pc group. \> LCSFactorTypes( G ) returns the abelian invariants of the lower central series factors of $G$. \> LCSFactorSizes( G ) returns the orders of the lower central series factors of $G$. \> WidthPGroup( G ) returns the width of $G$. \> SubgroupRank( G ) returns the (subgroup-)rank of $G$. \> Obliquity( G ) returns the obliquity of $G$. \> HasObliquityZero( G ) checks whether $G$ has obliquity 0 and returns true or false. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Functions to generate groups and trees} Let $G(p,rwo)$ denote the full tree of all finite $p$-groups with rank $rwo[1]$, width $rwo[2]$ and obliquity $rwo[3]$. This tree can be finite or infinite; if it is infinite, then the infinite pro-$p$-groups of the considered rank, width and obliquity specify infinite subtrees of the full tree. The groups not contained in such an infinite subtree are called sporadic. \> GroupsByRankWidthObliquity( p, d, rwo, roots, limit ) determines all $p$-groups $G$ with $G/\Phi(G)$ of order $p^d$ and rank, width and obliquity as prescribed in $rwo$ up to order $limit$. Here $p$ and $d$ are integers, $rwo$ is a list of three integers and $limit$ is an integer. The parameter $roots$ is a list of groups described by their id's with respect to the small groups library. The descendants of the groups described in $roots$ are excluded from the output of this function. This option can be used to prune the tree of groups determined by this function. If there are only finitely many sporadic $p$-groups with given rank, width and obliquity, then this function can be used to generate them; in this case $roots$ must contain a complete list of all id's of roots of infinite subtrees and $limit$ can be set to infinity. \> BranchRWO( G, i, rwo ) for a stable quotient (see \cite{ER09}) $G$ of a pro-$p$-group of rank $rwo[1]$, $rwo[2]$ and obliquity $rwo[3]$, this function returns the $i$-th branch of its corresponding tree. The structure of the tree is encoded in a list. If one of the global parameters `CHECK_RANK' or `CHECK_OBLIQUITY' is set to false, then checking the corresponding invariant is omitted and hence a potentially larger tree is returned. The user is advised not to perform any other computations using ANUPQ or the `pq'-program while using this or the following function, because such computations will be terminated. \> BoundedDescendantsRWO( G, i, c, rwo) returns the tree of all descendants of $G/\gamma_i(G)$ of rank $rwo[1]$, width $rwo[2]$, obliquity $rwo[3]$ and class at most $c$. \> DrawBranch( branch ) if the package is run under XGap, then this function can be used to draw a branch as output by the above two functions in the case of width 2. The user may wish to improve the quality of the output by modifying the file `gap/xbranch.gi'. Vertices drawn on the same level correspond to groups of the same class. If $G$ is a descendant of $H$ in the branch, then $G$ is drawn as a filled circle if $\vert G\vert = \vert H\vert p$ and as a solid box if $\vert G\vert = \vert H\vert p^2$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\Section{Quotients of pro-$p$-groups} The package also provides finite quotients of a number of infinite pro-$p$-groups with finite rank, width and obliquity. Throughout the section, $p$ is an odd prime. \> ProPSylowGroupOfPSL(d, p, n) returns the quotient of the Sylow pro-$p$-subgroup of ${\rm PSL}_d({\Q}_p)$ modulo the matrices which are congruent to the identity modulo $p^n$. \> ProPSylowGroupOfPSF(p, n) Let $L$ be the simple Lie algebra of dimension $3$ over $\Q_p$ which is not isomorphic to $sl_2(\Q_p)$. This function returns a finite quotient of the Sylow pro-$p$-subgroup of its automorphism group. The parameter $n$ specifies how large this quotient is. In \cite{KLGP97}, a library of maximal pro-$p$-groups with finite rank, width and obliquity corresponding to the Lie algebras of small dimension is provided. Here, we provide a library of large quotients of these groups for some of the Lie algebras of type $sl_d(K)$, where $K$ is a finite extension of $\Q_p$. These groups have been determined using the programs described in \cite{KLGP97}. To be precise, depending on the group, it may be necessary to pass to the quotient by one of the last non-trivial terms of the lower central series in order to obtain a quotient of the respective pro-$p$-group. \> ProPQuotient(p, dim, deg, no) returns a finite group corresponding to the maximal pro-$p$-group $G$ with Lie algebra $sl_{dim}(K)$, where $K$ is a field of degree $deg$ over $\Q_p$. The parameter $no$ specifies the number of the group in our database. \Section{Example} When run under XGap, the following code constructs and draws the branch with root $G/\gamma_5(G)$ in the graph of finite 5-groups of rank 3, width 2 and obliquity 0, where $G$ is the Sylow pro-$p$-subgroup of ${\rm Aut}(sl_2(\Q_5))$. \beginexample gap> g := ProPSylowGroupOfPSL(2,5,6); Pcp-group with orders [ 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5 ] gap> branch := BranchRWO(g,5,[3,2,0]);; ConstructBranch: root-p-class: 4 Constructed 3 1-step descendants. ConstructBranch: root-p-class: 5 Constructed 0 1-step descendants. Constructed 0 2-step descendants. ConstructBranch: root-p-class: 5 Constructed 0 1-step descendants. Constructed 0 2-step descendants. ConstructBranch: root-p-class: 5 Constructed 0 1-step descendants. Constructed 0 2-step descendants. Constructed 3 2-step descendants. ConstructBranch: root-p-class: 5 Constructed 0 1-step descendants. Constructed 0 2-step descendants. ConstructBranch: root-p-class: 5 Constructed 0 1-step descendants. Constructed 0 2-step descendants. gap> DrawBranch(branch); \endexample A window with the following graph should appear. \epsfxsize=8cm \epsfbox{branch.ps} ¤ Dauer der Verarbeitung: 0.25 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28