TY - JOUR

T1 - Torsion-free crystallographic groups with indecomposable holonomy group. II

AU - Bovdi, V. A.

AU - Gudivok, P. M.

AU - Rudko, V. P.

N1 - Funding Information:
* This research was supported by OTKA No. T 037202, No. T 038059 and No. T 034530.

PY - 2004

Y1 - 2004

N2 - 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.

AB - 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.

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U2 - 10.1515/jgth.2004.7.4.555

DO - 10.1515/jgth.2004.7.4.555

M3 - Article

AN - SCOPUS:6344292564

SN - 1433-5883

VL - 7

SP - 555

EP - 569

JO - Journal of Group Theory

JF - Journal of Group Theory

IS - 4

ER -