Periodic subgroups of projective linear groups in positive characteristic

We classify the maximal irreducible periodic subgroups of PGL(q, ), where is a field of positive characteristic p transcendental over its prime subfield, q = p is prime, and ^× has an element of order q. That is, we construct a list of irreducible subgroups G of GL(q, ) containing the centre ^×1 _q of GL(q, ), such that G/ ^×1 _q is a maximal periodic subgroup of PGL(q, ), and if H is another group of this kind then H is GL(q, )-conjugate to a group in the list. We give criteria for determining when two listed groups are conjugate, and show that a maximal irreducible periodic subgroup of PGL(q, ) is self-normalising.      

Recognition by spectrum for finite linear groups over fields of characteristic 2

The spectrum of a finite group is the set of its element orders. For every finite simple linear group L = L_n(2^k), where 11 ⩽ n ⩽ 18 or n > 24, we describe finite groups having the same spectrum as L, prove that the number of pairwise nonisomorphic groups with this property is finite, and derive an explicit formula for calculating this number.     

Normal-convexity and equations over groups

Summary A subgroupS of a groupH is said to be normal-convex inH if for any subsetRS, the natural mapS/R _S H/R _H is injective.In this paper, topological methods are used to show that normal-convexity is preserved under taking free products. In other words, ifS is normal-convex inH and ifT is normal-convex inK, thenS*T is normal-convex inH*K. Similar results are obtained for free products with amalgamation andHNN extensions. The method of proof uses a concept of normal-convexity defined for pairs of topological spaces.These results and the topological methods are applied to study the question of when a set of equations over a group has a solution in some overgroup. Equations over groups are defined in the following fashion. An equation over a groupH is of the formw=1 wherewH*F,F being some free groups, with its generators called theunknowns. The elements ofH appearing inw are called thecoefficients. The equationw=1 overH can be solved overH if there is a groupH ₁ containingH and possessing elements which satisfy the equationw=1 when substituted in for the unknowns.To any set of equations over a group, we associate a two-complex. The manner is analogous to that for presentations. The one-cells correspond to the unknowns, and the two-cells are attached according to the words obtained by ignoring the coefficients. The two-complex so constructed does not change when the coefficients or the groupH is changed. Thus different sets of equations may give rise to the same two-complex. We call a two-complexKervaire if any set of equations associated to it has a solution. Using the topological notion of normal-convexity, we show that the property of being Kervaire is preserved under subdivision, so in particular, it does not depend on the cell structure. Further, we show that the class of Kervaire complexes is closed under combinatorial extensions, connected-sum, cellular two-moves, and amalgamations along two-sided ₁-injective subcomplexes.     

On Galois cohomology of semisimple groups over local and global fields of positive characteristic

We treat a case that was omitted from consideration in our article [2] in Math Zeit, 2007.     

Computing invariants of reductive groups in positive characteristic

This paper gives an algorithm for computing invariant rings of reductive groups in arbitrary characteristic. Previously, only algorithms for linearly reductive groups and for finite groups have been known. The key step is to find a separating set of invariants.     

Deciding finiteness of matrix groups in positive characteristic

We present a new algorithm to decide finiteness of matrix groups defined over a field of positive characteristic. Together with previous work for groups in zero characteristic, this provides the first complete solution of the finiteness problem for finitely generated matrix groups over a field. We also give an algorithm to compute the order of a finite matrix group over a function field of positive characteristic by constructing an isomorphic copy of the group over a finite field. Our implementations of these algorithms are publicly available in Magma.     

On Galois cohomology of semisimple groups over local and global fields of positive characteristic, II

We show that the recent results of Prasad and Rapinchuk (Adv. Math. 207(2), 646–660, 2006) on the existence and uniqueness of certain global forms of semisimple algebraic groups with given local behaviour in the case of number fields still hold in the case of global function fields.     

On Galois cohomology of semisimple groups over local and global fields of positive characteristic, III

We extend some well-known results on Galois cohomology in its relation with weak approximation for connected linear algebraic groups over number fields to the case of global fields of positive characteristic. Some applications are considered.     

Stability radii of positive linear Volterra-Stieltjes equations

We study stability radii of linear Volterra-Stieltjes equations under multi-perturbations and affine perturbations. A lower and upper bound for the complex stability radius with respect to multi-perturbations are given. Furthermore, in some special cases concerning the structure matrices, the complex stability radius can precisely be computed via the associated transfer functions. Then, the class of positive linear Volterra-Stieltjes equations is studied in detail. It is shown that for this class, complex, real and positive stability radius under multi-perturbations or multi-affine perturbations coincide and can be computed by simple formulae expressed in terms of the system matrices. As direct consequences of the obtained results, we get some results on robust stability of positive linear integro-differential equations and of positive linear functional differential equations. To the best of our knowledge, most of the results of this paper are new.     

Representations of finite special linear groups in non-defining characteristic

We classify irreducible modules over the finite special linear group SLn(q) in the non-defining characteristic ?, describe restrictions of irreducible modules from GLn(q) to SLn(q), classify complex irreducible characters of SLn(q) irreducible modulo l, and discuss unitriangularity of the l-decomposition matrix for SLn(q).     

Soluble linear groups over Euclidean rings

We study soluble matrix groups over commutative rings and transfer some results of the theory of soluble matrix groups over a field to the case of soluble matrix groups over commutative rings.