Integral group ring of the Mathieu simple group M 12

We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic groupM ₁₂. As a consequence, we confirm for this group the Kimmerle’s conjecture on prime graphs. The research was supported by OTKA grants No. T 43034, No.K61007 and Francqui Stichting (Belgium) grant ADSI107.     

A representation of the Mathieu groupM 24 as a collineation group

Summary The Mathieu group M₂₄ can be represented as a collineation group in space of 11 dimensions over the field of two elements. This paper discusses the geometry associated with this representation. In memory of Guido Castelnuovo, in the recurrence of the first centenary of his birth.     

Integral group ring of Rudvalis 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 Rudvalis sporadic simple group Ru . As a consequence, for this group we confirm the Kimmerle conjecture on prime graphs. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 3–13, January, 2009.     

The Mathieu group M12

Let G be a finite simple non-Abelian group. t is an involution of G, and L=O² (C_G(t)/O · (C_G(t))). K the center Z(L) is cyclic and L/Z(L)-PGL (2, q), q odd, then either a Sylow 2-subgroup of G is semidihedral or C_G(t)Z₂×PGL (2.5) and G is isomorphic to the Mathieu group M₁₂ of degree 12.     

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 Kimmerle's conjecture on prime graphs.     

Minimal subschemes of the group association schemes of Mathieu groups

Minimal subschemes of the group association schemes of Mathieu groupsM _n (n = 11, 12, 22, 23, 24) are determined. It is proved that for eachM _n (n = 11, 12, 22, 23, 24), there is a unique minimal subscheme of. The class numbers of these minimal subschemes are 7, 11, 9, 11 and 20 respectively. A general computer program to determine subschemes of group association schemes of relatively small class numbers is discussd.     

Constructions of rank five geometries for the Mathieu group M 22

We construct nine rank five incidence geometries that are firm and residually connected and on which the Mathieu group M²² acts flag-transitively. The constructions use mainly objects arising from the Steiner systemS(3, 6, 22). One of these geometries was constructed by Meixner and Pasini in [10]. Three of them are obtained from the geometry of Meixner and Pasini using doubling (see [8] or [12]) or similar constructions. The remaining five are new and four of them have a star diagram. These latter four geometries are constructed using special partitions of the 22 points of the Steiner system S(3, 6, 22).     

Integral ring extensions and lo-pairs

We give an example of an LO-pair (R,T) with dim R=dimT = n, so that all maximal ideals of R have the same height n, but with T not integral over R. This answers an open question of David E. Dobbs.     

Nilpotency of the multiplicative group of a group ring

It is proven that if K is a commutative ring of characteristic p^m while group G contains p-elements, then the multiplicative group UKG of group ring KG is nilpotent if and only if G is nilpotent and its commutant G is a finite p-group. Those group algebras KG are described for which the nilpotency classes of groups G and UKG coincide.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 191–200, February, 1972.In conclusion, the author wishes to express her gratitude to A. A. Bovdi for his scientific direction.