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 -