Representations of finite posets and near-isomorphism of finite rank Butler groups
type: Item
instance of: scholarly article
title: Representations of finite posets and near-isomorphism of finite rank Butler groups
zbMATH Open document ID: 0835.20073
author: Manfred Dugas
author: David M. Arnold
published in: Rocky Mountain Journal of Mathematics
publication date: 1996-01-31T00:00:00Z
review text: The main theorem of this paper defines an additive functor \(F:B(T_n, p^m)\to\text{Rep}_{n+1}(Z/p^m Z)\) from categories of Butler groups to categories of finite abelian \(p\)-groups with distinguished subgroups. Let \(T_n=\{\tau_0,\tau_1,\dots,\tau_n\}\) be a set of types such that \(\tau_1,\dots,\tau_n\) are pairwise incomparable and the infimum of any two is \(\tau_0\): \(\tau_i\wedge\tau_j=\tau_0\) for \(1\leq i\neq j\leq n\). If \(p\) is a prime, \(B(T_n, p^m)\) denotes the category of Butler groups \(G\) with typeset contained in \(T_n\) that satisfy \(p^m G\subseteq\sum^n_{i=1} G(\tau_i)\). Morphisms are the usual homomorphisms. The category \(\text{Rep}_{n+1} (Z/p^m Z)\) has as objects \(n+2\)-tuples \((U, U_0, U_1,\dots, U_n)\), where \(U\) is a free \(Z/p^m Z\)-module and the \(U_i\) are submodules of \(U\). The morphisms are homomorphisms that preserve the distinguished submodules \(U_i\). Under the functor \(F\), near isomorphism classes of Butler groups are mapped to isomorphism classes of representations. Arguments and examples for almost completely decomposable (acd) groups are often easier to construct in \(\text{Rep}_{n+1} (Z/p^m Z)\) than in \(B(T_n, p^m)\). To illustrate this point of view, the authors use representations to construct, for \(p\neq 2\), indecomposable acd groups of arbitrarily large finite rank in \(B(T_n, p^m)\) for \(n\geq 3\) and sufficiently large \(m\); and in \(B(T_n, p)\) for \(n\geq 4\). The class \(B(T_3)\) of Butler groups with typeset in \(T_3\) is given special attention. If \(\tau_1\vee\tau_2\vee\tau_3=\text{type} (Q)\), then each acd group in \(B(T_3)\) is the direct sum of indecomposables of rank \(\leq 3\). A complete set of near isomorphism invariants is given for these groups. Using a complete list of indecomposables in \(\text{Rep}_4 (Z/pZ)\) developed by Gelfand- Ponomarev and later Brenner, the authors give a description of the indecomposables in \(B(T_3, p)\). The uniform (in the sense of Dugas- Oxford) acd gr…
Mathematics Subject Classification ID: 20K15
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 20K40
Mathematics Subject Classification ID: 20K99
zbMATH DE Number: 796891
zbMATH Keywords: indecomposable groups
zbMATH Keywords: almost completely decomposable groups
zbMATH Keywords: categories of Butler groups
zbMATH Keywords: finite Abelian \(p\)-groups
zbMATH Keywords: types
zbMATH Keywords: typesets
zbMATH Keywords: near isomorphism classes of Butler groups
zbMATH Keywords: isomorphism classes of representations
zbMATH Keywords: direct sums of indecomposables
zbMATH Keywords: near isomorphism invariants
reviewed by: Charles Vinsonhaler
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072239
OpenAlex ID: W2042074125
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DOI: 10.1216/RMJM/1181072239
description: scientific article; zbMATH DE number 796891