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We solve the following problem related to the Kneser-Tits conjecture, for Azumaya algebras. Given an Azumaya algebra D of rank 4 that is not a division algebra, whose center K is three-dimensional over the ground field F, such that corK/FD is trivial, is it true that every element of D having reduced norm in F is a product of n elements having both reduced norm and reduced trace in F? This is true for n ≥ 3, but false for n = 2. [ABSTRACT FROM AUTHOR]

In this paper we study special Moufang sets ( U, τ), with U abelian, under the additional restriction that they have finite Morley rank. Our result states that the little projective group of such a Moufang set must be isomorphic to PSL2( K) for an algebraically closed field K provided that U has characteristic 2 and that infinitely many endomorphisms of U centralize the Hua subgroup. This complements a result of De Medts and Tent that addresses the odd characteristic case. [ABSTRACT FROM AUTHOR]

The article provides a characterization of the Harada-Norton (HN) sporadic simple group as a bicharacteristic {2,5}. It is noted that many of the sporadic simple groups appear to have at least two characteristics, while there are several sporadic groups which should be regarded as groups of bicharacteristic {p,q} with {p,q} ≠ {2,3}. According to the article, a finite Group G K is local if every simple section of every local subgroup of G is a known finite simple group.

In this note we will give an elementary proof of the existence of sharply transitive R-modules M over principal ideal domains R. An R-module is sharply transitive (or a UT-module) if its R-automorphism group acts sharply transitively on the pure elements of M. We will assume that M is torsion-free; thus pure elements are simply those elements divisible only by units of R in M. We provide examples of UT-modules of rank ⩽ , while the existence of UT-modules of rank ⩾ was shown recently in Göbel and Shelah [R. Göbel and S. Shelah. Uniquely transitive torsion-free abelian groups. In Rings, modules, algebras, and abelian groups, Lecture Notes in Pure and Applied Math. 236 (Marcel Dekker, 2004), pp. 271–290.] using the more complicated machinery of prediction principles. The existence of countable abelian UT-groups, which follows from this note, was left open in earlier works. Here we require and exploit the existence of algebraically independent elements over the base ring R. (Thus we will need | R| < .) First we will convert the UT problem on modules (as suggested in Herden [D. Herden. Uniquely transitive R-modules. Ph.D. thesis. University of Duisburg-Essen, Campus Essen (2005).]) into a problem on suitable R-algebras. This reduces its solution to a few simple steps and makes the proofs more transparent, requiring only basic results in module theory. [ABSTRACT FROM AUTHOR]

The article discusses the extension of the characterization of the Harada-Norton sporadic simple group to bicharacteristic {2,p} for any prime p greater than 3. Based on the results, for such primes, a p-local configuration of the type found in the Harada-Norton group for p=5 can appear in a finite group only when p is equal to 3 or 5.

Abstract: We describe the action of idempotent transformations on finite groups. We show that finiteness is preserved by such transformations and enumerate all possible values such transformations can assign to a fixed finite simple group. This is done in terms of the first two homology groups. We prove for example that except special linear groups, such an orbit can have at most 7 elements. We also study the action of monomials of idempotent transformations on finite groups and show for example that orbits of this action are always finite. [Copyright &y& Elsevier]

Abstract: This paper provides a comprehensive investigation of the cellular approximation functor , in the category of groups, approximating a group G by a group A. We also study related notions such as A-injection, A-generation and A-constructibility of a group G and we find several interesting connections with the Schur multiplier . Our constructions are direct and are given in a slow and detailed manner. [Copyright &y& Elsevier]