An inequality in integral representation theory
Let p be a prime number and G an abelian group of order \(p^ n\), \(n\geq 1\). Let M be a module for the integral group ring ZG which is Z-free and finitely generated and which affords a faithful representation of G. Let V denote the corresponding QG-module, \(V=Q\otimes M=QM\). The main result of this paper gives a relation involving the \(rank_ Z(M)\), the dimension over \({\mathbb{F}}=Z/pZ\), of the space \((M/pM)^ G\) of G-fixed points on M/pM, and the elementary divisors of G. Let \(d_ i(G)\) denote the number of cyclic direct factors of G having order \(p^ i\) in a fixed decomposition of G as a direct product of cyclic groups, \(\phi\) is the usual Euler function, \(\lambda\) (V) is the composition length of the QG-module V. Theorem 1. Let G be a finite abelian p-group and let M be a Z-free ZG- module on which G acts faithfully. Then \[ rank_ Z(M)-\dim _{{\mathbb{F}}}(M/pM)^ G\geq \sum ^{e}_{i=1}d_ i(G)(\phi (p^ i)-1), \] where \(p^ e\) is the exponent of G. - The proof is carried out by proving two auxiliary results: Proposition. With G as above, and V any finite dimensional QG-module on which G acts faithfully, \[ \dim _ Q(V)\geq \sum ^{e}_{i=1}d_ i(G)\phi (p^ i). \] Proposition. Let M be any Z-free finitely generated ZG-module. Then \(\dim _{{\mathbb{F}}}(M/pM)^ G\leq \lambda (QM).\) The bound in Theorem 1 is best possible in the sense that for any given G, there is an M for which equality holds. Interest in bounds of this type is motivated by work of Wm. Browder and A. Adem in which they study groups acting on products of spheres.
scientific article; zbMATH DE number 4027731
scientific article