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 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.     

Zenkov Hypothesis on Pair p-Sylow Intersections of Chevalley Groups of Characteristic p

We present all Chevalley groups over a finite field of arbitrary characteristic p in which the index of the normalizer of any pair of Sylow p-subgroups is relatively prime with p.     

在其Sylow子群上作用双传递的有限群

本文完全确定了下述类型的有限群：（１）对于某个素数ｐ，在其Ｓｙｌｏｗｐ－子群的集合上作用双传递的所有几乎单群；（２）所有的非交换单群Ｔ，其自同构群在Ｔ的某个Ｓｙｌｏｗｐ－子群的集合上作用双传递；（３）在其所有的Ｓｙｌｏｗ子群上双传递的全部有限群．     

On Weakly Factorizable Groups

Groups with complemented subgroups, which are also called completely factorizable groups, were studied by P. Hall, S. N. Chernikov, and N. V. Chernikova (Baeva). For complete factorizability, it is sufficient (Theorem 1) that each proper subgroup have a normal complement in some larger subgroup. A group is said to be weakly factorizable if each of its proper subgroups is complemented in some larger subgroup; the problem of describing finite groups with this property is posed (Question 8.31) in the Kourovka Notebook. Some properties of these groups are considered. The question is studied for Sylow p-subgroups of Chevalley-type groups of characteristic p. The main theorem, Theorem 2, establishes the weak factorizability of the Sylow p-subgroups in the symmetric and alternative groups and in the classical linear groups over fields of characteristic p> 0, excluding the unitary groups of odd dimension > p.     

s-semipermutability and abnormality in finite groups *

A subgroup H of a group G is called s-semipermutable in G if it is permutable with all Sylow p-subgroups of G with (p,∣H∣) for all primes p such that p ∥G ∣. In this pa-per, we investigate the influence of s-semipermutable and abnormal subgroups on the structure of a finite group and classify such finite groups in which every subgroup is either s-semipermutable or abnormal.     

Rational division algebras as solvable crossed products

LetG be a finite group. If there exists a division algebra central over the rationalsQ which is a crossed product forG, then according to a theorem of Schacher, the Sylow subgroups ofG are all metacyclic. The converse is proved here to hold in the following cases: (1)G metacyclic; (2) The Sylow 2-subgroups ofG are cyclic (this impliesG solvable); (3)G is solvable and the Sylow 2-subgroups ofG are dihedral of order larger than 8.     

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.     

2-Signalizers of Finite Simple Groups

Maximal 2-signalizers and centralizers of Sylow 2-subgroups in all finite simple groups are described. Also normalizers are computed for Sylow 2-subgroups in the finite simple groups of exceptional Lie type over a field of odd characteristic.     

On <Emphasis Type="Italic">CAS</Emphasis>-subgroups of finite groups

We introduce a new subgroup embedding property of a finite group called CAS-subgroup. Using this subgroup property, we determine the structure of finite groups with some CAS-subgroups of Sylow subgroups. Our results unify and generalize some recent theorems on solvability, p-nilpotency and supersolvability of finite groups. The authors are supported by NSF of China (10571181) and NSF of Guangxi (0447038).     

On 2-sylow intersections

LetG be a finite group. We say thatG satisfiesI(k) if anyk+1 distinct Sylow 2-subgroups ofG interesect trivially. In this paper we classify all finite groups satisfyingI (3), and all finite simple groups satisfyingI (5).     

On Preservation of Stability for Finite Extensions of Abelian Groups

We characterize preservation of superstability and ω-stability for finite extensions of abelian groups and reduce the general case to the case of p-groups. In particular we study finite extensions of divisible abelian groups. We prove that superstable abelian-by-finite groups have only finitely many conjugacy classes of Sylow p-subgroups. Mathematics Subject Classification: 03C60, 20C05.     

Nearly m-embedded subgroups and solvability of finite groups

A subgroup A of a finite group G is called {1≤G}-embedded in G if for each two subgroups K≤H of G, where K is a maximal subgroup of H, A either covers the pair (K,H) or avoids it. Moreover, a subgroup H of G is called nearly m-embedded in G if G has a subgroup T and a {1≤G}-embedded subgroup C such that G?=?HT and H∩T≤C≤H. In this paper, we mainly prove that G is solvable if and only if its Sylow 3-subgroups, Sylow 5-subgroups and Sylow 7-subgroups are nearly m-embedded in G.     

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.     

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 a relation between the Sylow and Baer-Suzuki theorems

Given a set π of primes, say that a finite group G satisfies the Sylow π-theorem if every two maximal π-subgroups of G are conjugate; equivalently, the full analog of the Sylow theorem holds for π-subgroups. Say also that a finite group G satisfies the Baer-Suzuki π-theorem if every conjugacy class of G every pair of whose elements generate a π-subgroup itself generates a π-subgroup. In this article we prove, using the classification of finite simple groups, that if a finite group satisfies the Sylow π-theorem then it satisfies the Baer-Suzuki π-theorem as well.     

On the Pronormality of Subgroups of Odd Index in Some Extensions of Finite Groups

We study finite groups with the following property (*): All subgroups of odd index are pronormal. Suppose that G has a normal subgroup A with property (*), and the Sylow 2-subgroups of G/A are self-normalizing. We prove that G has property (*) if and only if so does N_G(T)/T, where T is a Sylow 2-subgroup of A. This leads to a few results that can be used for the classification of finite simple groups with property (*).     

Characters and Sylow 2-subgroups of maximal class revisited

We give two ways to distinguish from the character table of a finite group G if a Sylow 2-subgroup of G has maximal class. We also characterize finite groups with Sylow 3-subgroups of order 3 in terms of their principal 3-block.     

The Invariant Fields of the Sylow Groups of Classical Groups in the Natural Characteristic

Let X be any finite classical group defined over a finite field of characteristic p > 0. In this article, we determine the fields of rational invariants for the Sylow p-subgroups of X, acting on the natural module. In particular, we prove that these fields are generated by orbit products of variables and certain invariant polynomials which are images under Steenrod operations, applied to the respective invariant linear forms defining X.