V. A. Bovdi, P. M. Gudivok, V. P. Rudko
July 27, 2005
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 ℭ of M by G such that conjugation in ℭ induces the G -module structure on M . (When K = ℤ, these are just the classical crystallographic groups.) The dimension of ℭ is the K -rank of M , the holonomy group of ℭ is G , and ℭ 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 ℤ p of p -adic integers. We prove that the dimensions of such groups with G non-cyclic of order p 2 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.