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V. A. Bovdi

Publications and source records attributed to V. A. Bovdi.

31 records · Page 2Linked to original sources
arXiv2007

Unitary units in modular group algebras

Let p be a prime, K a field of characteristic p, G a locally finite p-group, KG the group algebra, and V the group of the units of KG with augmentation 1. The anti-automorphism g\mapsto g^{-1} of G extends linearly to KG; this extension leaves V setwise invariant, and its restriction to V followed by v\mapsto v^{-1} lives an automorphism of V. The elements of V fixed by this automorphism are called unitary; they form a subgroup. Our first theorem describes the K and G for which this subgroup is normal in V. For each element g in G, let \bar{g} denote the sum (in KG) of the distinct powers of g. The elements 1+(g-1)h\bar{g} with g,h\in G are the bicyclic units of KG. Our second theorem describes the K and G for which all bicyclic units are unitary.

math.RA↗
arXiv2007

Integral group ring of the McLaughlin simple group

We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the McLaughlin sporadic group McL. As a consequence, we confirm for this group the Kimmerle's conjecture on prime graphs.

math.RA↗
arXiv2007

Torsion units in integral group ring of Higman-Sims simple group

Using the Luthar-Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Higman-Sims simple sporadic group HS. As a consequence, we confirm the Kimmerle's conjecture on prime graphs for this sporadic group.

math.RA↗
arXiv2007

Torsion units in integral group ring of the Mathieu simple group M22

We investigate the possible character values of torsion units of the normalized unit group of the integral group ring of Mathieu sporadic group $M_{22}$. We confirm the Kimmerle conjecture on prime graphs for this group and specify the partial augmentations for possible counterexamples to the stronger Zassenhaus conjecture.

math.RA↗
arXiv2007

Integral group ring of the Mathieu simple group M24

We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic group $M_{24}$. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.

math.RA↗
arXiv2007

Integral group ring of the Mathieu simple group M23

We investigate the classical Zassenhaus conjecture for the unit group of the integral group ring of Mathieu simple group M23 using the Luthar-Passi method. This work is a continuation of the research that we carried out for Mathieu groups M11 and M12. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.

math.RA↗
arXiv2007

Integral group ring of the Mathieu simple group M12

We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic group M12. As a consequence, we confirm for this group the Kimmerle's conjecture on prime graphs.

math.RA↗
arXiv2007

Indecomposable linear groups

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.

math.RT↗
arXiv2006

Lie nilpotency indices of modular group algebras

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that if KG is Lie nilpotent then its upper (or lower) Lie nilpotency index is at most |G'|+1, where |G'| is the order of the commutator subgroup. The class of groups G for which these indices are maximal or almost maximal have already been determined. Here we determine G for which upper (or lower) Lie nilpotency index is the next highest possible.

math.RA↗
arXiv2006

Integral group ring of the first Mathieu simple group

We investigated the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the simple Mathieu group M11. As a consequence, for this group we confirm the conjecture by Kimmerle about prime graphs.

math.RA↗
arXiv2004

Torsion-free crystallographic groups with indecomposable holonomy group

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.

math.GR↗