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.
ASJC Scopus subject areas
- Algebra and Number Theory