Linear groups with rank restrictions on the subgroups of infinite central dimension

The authors study linear groups of infinite central dimension and of infinite p-rank all of whose proper subgroups of infinite p-rank are of finite central dimension.     

Locally soluble infinite-dimensional linear groups with restrictions on nonAbelian subgroups of infinite ranks

We are concerned with locally soluble linear groups of infinite central dimension and infinite sectional p-rank, p ⩾ 0, in which every proper non-Abelian subgroup of infinite sectional p-rank has finite central dimension. It is proved that such groups are soluble. Translated from Algebra i Logika, Vol. 47, No. 5, pp. 601–616, September–October, 2008.     

Infinite-dimensional linear groups with restrictions on subgroups that are not soluble <Emphasis Type="Italic">A</Emphasis><Subscript>3</Subscript>-groups

We are concerned with infinite-dimensional locally soluble linear groups of infinite central dimension that are not soluble A₃-groups and all of whose proper subgroups, which are not soluble A₃-groups, have finite central dimension. The structure of groups in this class is described. The case of infinite-dimensional locally nilpotent linear groups satisfying the specified conditions is treated separately. A similar problem is solved for infinite-dimensional locally soluble linear groups of infinite fundamental dimension that are not soluble A₃-groups and all of whose proper subgroups, which are not soluble A₃-groups, have finite fundamental dimension. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 548–559, September–October, 2007.     

Torsion-free constructive nilpotent <Emphasis Type="Italic">R</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-groups

We consider a torsion-free nilpotent R _p -group, the p-rank of whose quotient by the commutant is equal to 1 and either the rank of the center by the commutant is infinite or the rank of the group by the commutant is finite. We prove that the group is constructivizable if and only if it is isomorphic to the central extension of some divisible torsion-free constructive abelian group by some torsion-free constructive abelian R _p -group with a computably enumerable basis and a computable system of commutators. We obtain similar criteria for groups of that type as well as divisible groups to be positively defined. We also obtain sufficient conditions for the constructivizability of positively defined groups.     

On modules over integer-valued group rings of locally soluble groups with rank restrictions imposed on subgroups

We study a \mathbbZG \mathbb{Z}G -module A such that \mathbbZ \mathbb{Z} is the ring of integer numbers, the group G has an infinite sectional p-rank (or an infinite 0-rank), C _G (A) = 1, A is not a minimax \mathbbZ \mathbb{Z} -module, and, for any proper subgroup H of infinite sectional p-rank (or infinite 0-rank, respectively), the quotient module A/C _A (H) is a minimax \mathbbZ \mathbb{Z} -module. It is shown that if the group G is locally soluble, then it is soluble. Some properties of soluble groups of this kind are discussed.     

On locally primitive Cayley graphs of finite simple groups

In this paper we investigate locally primitive Cayley graphs of finite nonabelian simple groups. First, we prove that, for any valency d for which the Weiss conjecture holds (for example, d?20 or d is a prime number by Conder, Li and Praeger (2000) [1]), there exists a finite list of groups such that if G is a finite nonabelian simple group not in this list, then every locally primitive Cayley graph of valency d on G is normal. Next we construct an infinite family of p-valent non-normal locally primitive Cayley graph of the alternating group for all prime p?5. Finally, we consider locally primitive Cayley graphs of finite simple groups with valency 5 and determine all possible candidates of finite nonabelian simple groups G such that the Cayley graph Cay(G,S) might be non-normal.     

Modules over group rings of solvable groups with rank restrictions on subgroups

Let A be an R G-module over a commutative ring R, where G is a group of infinite section p-rank (0-rank), C _G (A) = 1, A is not a Noetherian R-module, and the quotient A/C _A (H) is a Noetherian R-module for every proper subgroup H of infinite section p-rank (0-rank). We describe the structure of solvable groups G of this type.     

The p-rank stratification on the Siegel moduli space with Iwahori level structure

Our concern in this paper is to describe the p-rank stratification on the Siegel moduli space with Iwahori level structure over fields of positive characteristic. We calculate the dimension of the strata and describe the closure of a given stratum in terms of p-rank strata. We also examine the relationship between the p-rank stratification and the Kottwitz–Rapoport stratification.     

On recognition of finite simple groups with connected prime graph

The spectrum of a finite group is the set of its element orders. We prove a theorem on the structure of a finite group whose spectrum is equal to the spectrum of a finite nonabelian simple group. The theorem can be applied to solving the problem of recognizability of finite simple groups by spectrum.     

On Modular Standard Modules of Association Schemes

We will determine the structure of the modular standard modules of association schemes of class two. In the process, we will give the theoretical interpretation for the p-rank theory for strongly regular graphs, and understand the p-rank as the dimension of a submodule of the modular standard module. Considering the modular standard module, we can obtain the detailed classification more than the p-rank and the parameters.     

Affine geometry designs, polarities, and Hamada's conjecture

In a recent paper, two of the authors used polarities in PG(2d−1,p) (p?2 prime, d?2) to construct non-geometric designs having the same parameters and the same p-rank as the geometric design PGd(2d,p) having as blocks the d-subspaces in the projective space PG(2d,p), hence providing the first known infinite family of examples where projective geometry designs are not characterized by their p-rank, as it is the case in all known proven cases of Hamada's conjecture. In this paper, the construction based on polarities is extended to produce designs having the same parameters, intersection numbers, and 2-rank as the geometric design AGd+1(2d+1,2) of the (d+1)-subspaces in the binary affine geometry AG(2d+1,2). These designs generalize one of the four non-geometric self-orthogonal 3-(32,8,7) designs of 2-rank 16 (V.D. Tonchev, 1986 [12]), and provide the only known infinite family of examples where affine geometry designs are not characterized by their rank.     

The permutation action of finite symplectic groups of odd characteristic on their standard modules

We study the space of functions on a finite-dimensional vector space over a field of odd order as a module for a symplectic group. We construct a basis of this module with the following special properties. Each submodule generated by a single basis element under the symplectic group action is spanned as a vector space by a subset of the basis and has a unique maximal submodule. From these properties, the dimension and composition factors of the submodule generated by any subset of the basis can be determined. These results apply to incidence geometry of the symplectic polar space, yielding the symplectic analogue of Hamada's additive formula for the p-ranks of the incidence matrices between points and flats. A special case leads to a closed formula for the p-rank of the incidence matrix between the points and lines of the symplectic generalized quadrangle over a field of odd order. Together with earlier results on the 2-ranks, this result completes the determination of the p-ranks for these quadrangles.     

Exponent and p-rank of finite p-groups and applications

We bound the order of a finite p-group in terms of its exponent and p-rank. Here the p-rank is the maximal rank of an abelian subgroup. These results are applied to defect groups of p-blocks of finite groups with given Loewy length. Doing so, we improve results in a recent paper by Koshitani, Külshammer, and Sambale. In particular, we determine possible defect groups for blocks with Loewy length 4.     

Asymptotically good towers of function fields with small p-rank

Over any quadratic finite field we construct function fields of large genus that have simultaneously many rational places, small p-rank, and many automorphisms.     

Minimum product set sizes in nonabelian groups of order pq

Let G be a nonabelian group of order pq, where p and q are distinct odd primes. We analyze the minimum product set cardinality μG(r,s)=min|AB|, where A and B range over all subsets of G of cardinalities r and s, respectively. In this paper, we completely determine μG(r,s) in the case where G has order 3p and conjecture that this result can be extended to all nonabelian groups of order pq. We also prove that for every nonabelian group of order pq there exist 1?r,s?pq such that μG(r,s)>μZ/pqZ(r,s).     

On one class of modules that are close to Noetherian

We consider an R G-module A over a commutative Noetherian ring R. Let G be a group having infinite section p-rank (or infinite 0-rank) such that C _G (A) = 1, A/C _A (G) is not a Noetherian R-module, but the quotient A/C _A (H) is a Noetherian R-module for every proper subgroup H of infinite section p-rank (or infinite 0-rank, respectively). In this paper, it is proved that if G is a locally soluble group, then G is soluble. Some properties of soluble groups of this type are also obtained.     

The p-Rank of Ramified Covers of Curves

In this paper we study the p-rank of Abelian prime-to-p covers of the generic r-pointed curve of genus g. There is an obvious bound on the p-rank of the cover. We show that it suffices to compute the p-rank of cyclic prime-to-p covers of the generic r-pointed curve of genus zero. In that situation, we show that, for large p, the p-rank of the cover is equal to the bound.     

Finite groups with planar subgroup lattices

It is natural to ask when a group has a planar Hasse lattice or more generally when its subgroup graph is planar. In this paper, we completely answer this question for finite groups. We analyze abelian groups, p-groups, solvable groups, and nonsolvable groups in turn. We find seven infinite families (four depending on two parameters, one on three, two on four), and three “sporadic” groups. In particular, we show that no nonabelian group whose order has three distinct prime factors can be planar.     

On some ranks of infinite groups

Abstract A group G has finite Hirsch-Zaicev rank r_hz(G) = r if G has an ascending series whose factors are either infinite cyclic or periodic and if the number of infinite cyclic factors is exactly r. The authors discuss groups with finite Hirsch-Zaicev rank and the connection between this and groups having finite section p-rank for some prime p, or p=0. Groups all of whose abelian subgroups are of bounded rank are also discussed. Keywords: p-rank, locally generalized radical group, Hirsch-Zaicev rank, torsion-free rank, rank Mathematics Subject Classification (2000): 20F19, 20E25, 20E15     

Thompson’s conjecture for alternating group of degree 22

For a finite group G, it is denoted by N(G) the set of conjugacy class sizes of G. In 1980s, J. G. Thompson posed the following conjecture: if L is a finite nonabelian simple group, G is a finite group with trivial center, and N(G) = N(L), then L and G are isomorphic. In this paper, it is proved that Thompson’s conjecture is true for the alternating group A ₂₂ with connected prime graph.