The derived p-length of finite p-soluble groups

The notion of derived p-length of a finite p-soluble group is introduced and its elementary properties are described. New estimates of the derived p-length of a finite p-soluble group are obtained depending on the construction of Sylow p-subgroups.     

On fixed point sets of distinguished collections for groups of parabolic characteristic

We determine the nature of the fixed point sets of groups of order p, acting on complexes of distinguished p-subgroups (those p-subgroups containing p-central elements in their centers). The case when G has parabolic characteristic p is analyzed in detail.     

The finite groups with no real p-elements

Given a prime p, we investigate the finite groups with no nontrivial real p-elements. Under certain natural hypotheses, we show that these groups have abelian Sylow p-subgroups.     

On second maximal subgroups of Sylow subgroups of finite groups

Finite groups in which the second maximal subgroups of the Sylow p-subgroups, p a fixed prime, cover or avoid the chief factors of some of its chief series are completely classified.     

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.     

<Emphasis Type="Italic">p</Emphasis>-Subgroups in Automorphism Groups of Real del Pezzo Surfaces

We classify finite p-groups acting minimally on real del Pezzo surfaces. This gives a partial classification of p-subgroups in the real plane Cremona 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.     

Characterization of TL-uniform spaces by coverings

In this paper we introduce the concepts of fuzzy TL-uniform spaces by means of coverings, where T stands for any continuous triangular norm. We show that the structure of covering TL-uniform spaces are isomorphic to fuzzy TL-uniform spaces as defined by Hashem and Morsi (2006) [4]. In particular, we study the continuity of functions between covering TL-uniform spaces, the I-topological space associated with a covering TL-uniform space. Also, we define the notions of level covering uniformities for a covering TL-uniformity. Moreover, we deduce a number of functors between categories of covering TL-uniform spaces, fuzzy TL-uniform spaces, covering uniform spaces and uniform spaces.     

On the regular Sylow p-subgroups of Chevalley groups over \mathbb{Z}_{p^m }

We prove that a Sylow p-subgroup of the general linear group of dimension n over the residue ring modulo p ^m is regular for n ² < p and powerful if and only if n = 2 and m = 1. We obtain similar results for the Sylow p-subgroups of normal types over the same ring.     

OnTL-finite state machines

In this paper we introduce the notions of aTL-finite state machine,TL-retrievability,TL-separability,TL-connectivity and discuss their basic properties.     

A theorem of Jon F. Carlson on filtrations of modules

We give an alternative proof to a theorem of Carlson [J.F. Carlson, Cohomology and induction from elementary abelian subgroups, Quart. J. Math. 51 (2000) 169-181] which states that if G is a finite group and k is a field of characteristic p, then any kG-module is a direct summand of a module which has a filtration whose sections are induced from elementary abelian p-subgroups of G. We also prove two new theorems which are closely related to Carlson’s theorem.     

New Collections of p-Subgroups and Homology Decompositions for Classifying Spaces of Finite Groups

We define new collections of p-subgroups for a finite group G and p a prime dividing its order. We study the homotopy relations among them and with the standard collections of p-subgroups and determine their ampleness and sharpness properties.     

On <Emphasis Type="Italic">π</Emphasis>-S-permutable subgroups of finite groups

Let π be a set of primes. A subgroup H of a finite group G is said to be π-S-permutable in G if H permutes with every Sylow q-subgroup of G for all primes q ∈ π. The main aim of this paper is to establish structural results about the normal closure of π-S-permutable subgroups and p-subgroups permuting with all p′-subgroups for a single prime p. Our results stem from a recent article by Isaacs [5] and subsequent discussions with the authors about it.     

Recognizing Abelian Sylow Subgroups in Finite Groups

Let p be a prime. We prove that if a finite group G has non-abelian Sylow p-subgroups, and the class size of every p-element in G is coprime to p, then G contains a simple group as a subquotient which exhibits the same property. In addition, we provide a list of all the simple groups and primes such that the Sylow p-subgroups are non-abelian and all p-elements have class size coprime to p.     

Finite groups with three conjugacy class sizes of some elements

Let G be a finite group. We prove as follows: Let G be a p-solvable group for a fixed prime p. If the conjugacy class sizes of all elements of primary and biprimary orders of G are {1,p ^a , n} with a and n two positive integers and (p,n)?=?1, then G is p-nilpotent or G has abelian Sylow p-subgroups.     

Some sufficient conditions for a finite group to be supersolvable

This paper represents an attempt to extend and improve the following result of Berkovich: Let G be a group of odd order. Let G=G ₁ G ₂ such that G ₁ and G ₂ are subgroups of?G. If the Sylow p-subgroups of G ₁ and of G ₂ are cyclic, then G is p-supersolvable.     

Towards the generalized purely wild inertia conjecture for product of alternating and symmetric groups

We obtain new evidence for the Purely Wild Inertia Conjecture posed by Abhyankar and for its generalization. We show that this generalized conjecture is true for any product of simple Alternating groups in odd characteristics, and for any product of certain Symmetric or Alternating groups in characteristic two. We also obtain important results towards the realization of the inertia groups which can be applied to more general set up. We further show that the Purely Wild Inertia Conjecture is true for any product of perfect quasi p-groups (groups generated by their Sylow p-subgroups) if the conjecture is established for individual groups.     

OnTL-subsystems ofTL-finite state machines

In this paper we introduce the notions ofTL-subsystems, strongTL-subsystems and discuss their basic properties.     

On K-ℙ-subnormal subgroups of finite groups

A subgroup H of a group G is said to be K-?-subnormal in G if H can be joined to the group by a chain of subgroups each of which is either normal in the next subgroup or of prime index in it. Properties of K-?-subnormal subgroups are obtained. A class of finite groups whose Sylow p-subgroups are K-?-subnormal in G for every p in a given set of primes is studied. Some products of K-?-subnormal subgroups are investigated.     

Classifying spaces of Kač-Moody groups

We study the structure of classifying spaces of Kač-Moody groups from a homotopy theoretic point of view. They behave in many respects as in the compact Lie group case. The mod p cohomology algebra is noetherian and Lannes'T functor computes the mod p cohomology of classifying spaces of centralizers of elementary abelian p-subgroups. Also, spaces of maps from classifying spaces of finite p-groups to classifying spaces of Kač-Moody groups are described in terms of classifying spaces of centralizers while the classifying space of a Kač-Moody group itself can be described as a homotopy colimit of classifying spaces of centralizers of elementary abelian p-subgroups, up to p-completion. We show that these properties are common to a larger class of groups, also including parabolic subgroups of Kač-Moody groups, and centralizers of finite p-subgroups. Received: 15 June 2000 / in final form: 20 September 2001 / Published online: 29 April 2002