Torsion-free crystallographic groups with indecomposable holonomy group. II

V. A. Bovdi, P. M. Gudivok, V. P. Rudko

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

Let K be a principal ideal domain, G a finite group, and M a KG-module which is a free K-module of finite rank on which G acts faithfully. A generalized crystallographic group is a non-split extension script C sign of M by G such that conjugation in script C sign induces the G-module structure on M. (When K = ℤ, these are just the classical crystallographic groups.) The dimension of script C sign is the K-rank of M, the holonomy group of script C sign is G, and script C sign is indecomposable if M is an indecomposable KG-module. We study indecomposable torsion-free generalized crystallographic groups with holonomy group G when K is ℤ, or its localization ℤ(p) at the prime p, or the ring ℤp of p-adic integers. We prove that the dimensions of such groups with G non-cyclic of order p2 are unbounded. For K = ℤ, we show that there are infinitely many non-isomorphic such groups with G the alternating group of degree 4 and we study the dimensions of such groups with G cyclic of certain orders.

Original languageEnglish
Pages (from-to)555-569
Number of pages15
JournalJournal of Group Theory
Volume7
Issue number4
DOIs
Publication statusPublished - 2004
Externally publishedYes

ASJC Scopus subject areas

  • Algebra and Number Theory

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