Representations of finite posets and near-isomorphism of finite rank Butler groups
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 groups in \(B(T_3, p^m)\) are shown to have finitely many near isomorphism classes for \(m=2\), but infinitely many for \(m\geq 4\). A number of open questions are posed.
near isomorphism classes of Butler groups
scientific article; zbMATH DE number 796891
scientific article