Victor Bovdi

ZASSENHAUS CONJECTURE FOR CENTRAL EXTENSIONS OF S5

Abstract We confirm a conjecture of Zassenhaus about rational conjugacy of torsion units in integral group rings for a covering group of the symmetric group S 5 and for the general linear group GL(2, 5). The first result, together with others from the literature, settles the conjugacy question for units of prime-power order in the integral group ring of a finite Frobenius group.

Torsion units in integral group rings of Janko simple groups

Using the Luthar-Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of integral group rings of Janko sporadic simple groups. As a consequence, we obtain that the Gruenberg-Kegel graph of the Janko groups J 1 , J 2 and J 3 is the same as that of the normalized unit group of their respective integral group ring.

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 groupM12. As a consequence, we confirm for this group the Kimmerle’s conjecture on prime graphs.

Groups St Andrews 2005: Integral group ring of the first Mathieu simple group

We investigate 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.

Integral Group Ring of the Mathieu Simple Group M 23

We investigate the classical Zassenhaus conjecture for the unit group of the integral group ring of Mathieu simple group M 23 using the Luthar–Passi method. This work is a continuation of the research that we carried out for Mathieu groups M 11 and M 12. As a consequence, for this group we confirm Kimmerles conjecture on prime graphs.

Symmetric units in modular group algebras

Let p be a prime, G a locally finite p-group, K a commutative ring of characteristic p. The anti-automorphism g\mapsto g\m1 of G extends linearly to an anti-automorphism a\mapsto a^* of KG. An element a of KG is called symmetric if a^*=a. In this paper we answer the question: for which G and K do the symmetric units of KG form a multiplicative group.

On the order of the unitary subgroup of a modular group algebra

Let KG be a group algebra of a finite p-group G over a finite field Kof characteristic p. We compute the order of the unitary subgroup of the group of units when G is either an extraspecial 2-group or the central product of such a group with a cyclic group of order 4 or G has an abelian subgroup A of index 2 and an element b such that b inverts each element of A.

On symmetric units in group algebras

Let U(KG) be the group of units of the group ring KG of the group G over a commutative ring K. The anti-automorphism g → g −1 of G can be extended linearly to an anti-automorphism a → a * of KG. Let S * (KG) = {x ∈ U(KG) | x * = x} be the set of all symmetric units of U(KG). We consider the following question: for which groups G and commutative rings K it is true that S * (KG) is a subgroup in U(KG). We answer this question when either a) G is torsion and K is a commutative G-favourable integral domain of characteristic p≥ 0 or b) G is non-torsion nilpotent group and KG is semiprime.

On a filtered multiplicative bases of group algebras II

We give an explicit list of all p-groups G with a cyclic subgroup of index p2, such that the group algebra KG over the field K of characteristic p has a filtered multiplicative K-basis. We also prove that such a K-basis does not exist for the group algebra KG, in the case when G is either a non-Abelian powerful p-group or a two generated p-group (p≠2) with a central cyclic commutator subgroup. This paper is a continuation of the paper which appeared in Arch. Math. (Basel)74 (2000), 217–285.

TORSION UNITS IN INTEGRAL GROUP RINGS OF CONWAY SIMPLE GROUPS

Using the Luthar–Passi method, we investigate the possible orders and partial augmentations of torsion units of the normalized unit group of integral group rings of Conway simple groups Co1, Co2 and Co3.