On the Influence of the Indices of Normalizers of Sylow Subgroups on the Structure of a Finite p-Soluble Group

We study the dependence of the structure of finite p-soluble groups on the indices of normalizers of Sylow subgroups. We obtain estimates for the p-length of these groups, and for small values of indices we find the nilpotent length of a soluble group.     

On finite groups with special Sylow <Emphasis Type="Italic">p</Emphasis>-subgroups

We find the cases in which a finite p-soluble group with a special Sylow p-subgroup has p-length 1.     

Finite groups in which all p-subnormal subgroups form a lattice

O. Kegel, in 1962, introduced the concept of p-subnormal subgroups of a finite group as the subgroups whose intersections with all Sylow p-subgroups of the group are Sylow p-subgroups of the subgroup. The set of p-subnormal subgroup of a finite group is not a lattice in general. In this paper, the class of all finite groups in which all p-subnormal subgroups from a lattice is determined. This is the class of all finite p-soluble groups whose p-length and p′-length, both, are less or equal to 1. The join-semilattice case and the meet-semilattice case are analyzed separately. The authors are supported by Proyecto PB 94-1048 of DGICYT, Ministerio de Educación y Ciencia of Spain.     

On the Influence of the Indices of Normalizers of Sylow Subgroups on the Structure of a Finite p-Soluble Group

We study the dependence of the structure of finite p-soluble groups on the indices of normalizers of Sylow subgroups. We obtain estimates for the p-length of these groups, and for small values of indices we find the nilpotent length of a soluble group.     

On the p-length of some finite p-soluble groups

The main aim of this paper is to give structural information of a finite group of minimal order belonging to a subgroup-closed class of finite groups and whose p-length is greater than 1, p a prime number. Alternative proofs and improvements of recent results about the influence of minimal p-subgroups on the p-nilpotence and p-length of a finite group arise as consequences of our study.     

On the $${{\varvec{}}}$$-length of the mutually ermutable roduct of two $${{\varvec{}}}$$-soluble grous

Let a finite group \(G=AB\) be the mutually permutable product of two p-soluble subgroups A and B for some prime p. We give a bound of the p-length of G from the p-lengths of A and B.     

The derived π-length, nilpotent π-length, and simple π-length of finite π-soluble groups

The concept of the derived π-length for finite π-soluble groups is introduced and its elementary properties are described. The dependence between the π-length, nilpotent π-length, and derived π-length, and also between the derived and nilpotent lengths of a π-Hall subgroup, is determined.     

On finite soluble groups of fixed rank

We establish the dependence of the derived length and p-length of a finite soluble group on its rank.     

A Characterisation of p-Soluble Groups

If p is a prime, then a finite group is p-soluble if each ofits composition factors is either a p-group or has order coprimeto p. For example, soluble groups are p-soluble. However, thereare many insoluble groups that are p-soluble. We shall provethe following result. 1991 Mathematics Subject Classification20D10.     

On the p-Length of a Product of Two Schmidt Groups

It is established that a finite p-solvable group presenting the product of two of its Schmidt subgroups has p-length at most 2.     

The nilpotent π-length of finite π-solvable groups with restrictions on subgroups

V.S. Monakhov’s conjecture concerning an estimate of the nilpotent π-length of a π-solvable group G is confirmed in the case l _p (G) ≤ 1 for all p ∈ π ? {q} and q ∈ π. New estimates of the nilpotent π-length of a π-solvable group with a supersolvable π-Hall subgroup are also obtained.     

Broué's conjecture for non-principal 3-blocks of finite groups

In representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group D, then A and its Brauer correspondent p-block B of N_G(D) are derived equivalent. We demonstrate in this paper that Broué's conjecture holds for two non-principal 3-blocks A with elementary abelian defect group D of order 9 of the O'Nan simple group and the Higman-Sims simple group. Moreover, we determine these two non-principal block algebras over a splitting field of characteristic 3 up to Morita equivalence.     

On lengths of <Emphasis Type="Italic">H</Emphasis>ℤ-localization towers

In this paper, the H?-length of different groups is studied. By definition, this is the length of the H?-localization tower or the length of the transfinite lower central series of H?-localization. It is proved that, for a free noncyclic group, its H?-length is ≥ ω+2. For a large class of ?[C]-modules M, where C is an infinite cyclic group, it is proved that the H?-length of the semi-direct product M ? C is ≤ ω + 1 and its H?-localization can be described as a central extension of its pro-nilpotent completion. In particular, this class covers modules M, such that M?C is finitely presented and H₂(M ? C) is finite.     

Linear closures of finite geometries

The interrelations between finite geometries (finite incidence structures) and linear codes over finite fields are discussed under some special fundamental aspects. For any incidence structure \({\mathcal{I}}\) block codes, block-difference codes and co-block codes over finite fields of characteristic p are discussed resp. introduced; correspondingly p-modular co-blocks are defined for \({\mathcal{I}}\). Orthogonality modulo p is introduced as a concept relating different geometries having the same point set. Conversely three types of block-tactical geometries may be derived from vector classes of fixed Hamming weight in a given linear code. These geometries are tactical configurations if the given code admits a transitive permutation group. A combination of both approaches leads to the concept of p-closure of a finite geometry and to the notions of p-closed, weakly p-closed and p-dense incidence structures. These geometric concepts are applied to simple or directed graphs via their natural “adjacency geometry”. Here the above mentioned code theoretic treatment leads to the concept of p-modular co-adjacent vertex sets. As instructive examples the Petersen graph, its complemetary graph and the Higman-Sims graph are considered.     

On the second commutants of finite Alperin groups

We refer to an Alperin group as a group in which the commutant of every 2-generated subgroup is cyclic. Alperin proved that if p is an odd prime then all finite p-groups with the property are metabelian. Nevertheless, finite Alperin 2-groups may fail to be metabelian. We prove that for each finite abelian group H there exists a finite Alperin group G for which G″ is isomorphic to H.     

Finite Groups with p-Multiplicative Probabilistic Zeta Function

We prove that a finite group G is p-soluble if and only if the Dirichlet polynomial P _G (s) is p-multiplicative. Moreover, we show that one can recover the set of prime divisors of the order of G from the knowledge of P _G (s).     

Character tables of p-groups with derived subgroup of prime order I

Two character tables of finite groups are isomorphic if there exist a bijection for the irreducible characters and a bijection for the conjugacy classes that preserve all the character values. We give necessary and sufficient conditions for two finite groups to have isomorphic character tables. In the case of finite p-groups with derived subgroup of order p, we show that the character tables can be classified by equivalence classes of certain homomorphisms of abelian p-groups.     

Groups with a positive law on commutators

Let p be a prime number and let G be a finitely generated group that is residually a finite p-group. We prove that if G satisfies a positive law on all elements of the form [a,b][c,d]ⁱ, a,b,c,d∈G and i?0, then the entire derived subgroup G^′ satisfies a positive law. In fact, G^′ is an extension of a nilpotent group by a locally finite group of finite exponent.     

Finite groups whose irreducible Brauer characters have prime power degrees

Let G be a finite group and let p be a prime. In this paper, we classify all finite quasisimple groups in which the degrees of all irreducible p-Brauer characters are prime powers. As an application, for a fixed odd prime p, we classify all finite nonsolvable groups with the above-mentioned property and having no nontrivial normal p-subgroups. Furthermore, for an arbitrary prime p, a complete classification of finite groups in which the degrees of all nonlinear irreducible p-Brauer characters are primes is also obtained.