arXiv subjects
Publications and source records attributed to V. P. Rudko.
Let G be a noncyclic group of order 4, and let K be the ring Z of rational integers, the localization of Z at the prime 2 and the ring of 2-adic integers, respectively. We describe, up to conjugacy, all of the indecomposable subgroups in the group GL(m,K) which are isomorphic to G.
For the ring R of integers of a ramified extension of the field of p-adic numbers and a cyclic group G of prime order p we study the extensions of the additive groups of R-representations modules of G by the group G.
Let K be a principal ideal domain, G a finite group, and M a KG-module which as K-module is free of finite rank, and on which $G$ acts faithfully. A generalized crystallographic group (introduced by the authors in volume 5 of Journal of Group Theory) is a group $\frak C$ which has a normal subgroup isomorphic to M with quotient G, such that conjugation in $\frak C$ gives the same action of G on M that we started with. (When $K=\Bbb Z$, these are just the classical crystallographic groups.) The K-free rank of M is said to be the dimension of $\frak C$, the holonomy group of $\frak C$ is G, and $\frak C$ is called indecomposable if M is an indecomposable KG-module. Let K be either $\Bbb Z$, or its localization $\Bbb Z_{(p)}$ at the prime p, or the ring $\Bbb Z_p$ of p-adic integers, and consider indecomposable torsionfree generalized crystallographic groups whose holonomy group is noncyclic of order p^2. In Theorem 2, we prove that (for any given p) the dimensions of these groups are not bounded. For $K=\Bbb Z$, we show in Theorem 3 that there are infinitely many non-isomorphic indecomposable torsionfree crystallographic groups with holonomy group the alternating group of degree 4. In Theorem 1, we look at a cyclic G whose order |G| satisfies the following condition: for all prime divisors p of |G|, p^2 also divides G, and for at least one p, even p^3 does. We prove that then every product of |G| with a positive integer coprime to it occurs as the dimension of some indecomposable torsionfree crystallographic group with holonomy group G.
We study the torsion free generalized crystallographic groups with the indecomposable holonomy group which is isomorphic to either a cyclic group of order ${p^s}$ or a direct product of two cyclic groups of order ${p}$.