The p-local ranks of finite simple groups with abelian Sylow p-subgroups

As a continuation of previous work, in this paper, we mainly study the finite simple groups which have abelian Sylow subgroups in term of p-local rank, especially a group theoretic characterization will be given.     

The order of elements in Sylow p-subgroups of the symmetric group

Define a random variable ξ_n by choosing a conjugacy class C of the Sylow p-subgroup of S_pn by random, and let ξ_n be the logarithm of the order of an element in C. We show that ξ_n has bounded variance and mean order log n /log p +O(1), which differs greatly from the average order of elements chosen with equal probability. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

有限交换环上极限线性群的Sylowp—子群

本文给出了有限交换局部环Ｒ上无限线性群ＧＬ（Ｒ）＝∪ｎＧＬｎＲ的Ｓｙｌｏｗｐ－子群的形式．令Ｍ是有限交换局部环Ｒ的唯一极大理想，ｋ＝Ｒ／Ｍ为Ｒ的剩余类域．用Ｘ（ｋ）表示ｋ的特征，并假定Ｐ与ｘ（ｋ）互素．作者证明了：ＧＬ（Ｒ）的任一Ｓｙｌｏｗｐ－子群Ｓ或者同构于的可数无限直积与Ｐ（ｊ）的无限直积的直积（当Ｐ≠２或Ｐ＝２，Ｘ（ｋ）β≡１（ｍｏｄ４））或者同构于Ｐｉ的无限直积与Ｐ（ｊ）的无限直积的直积（当Ｐ＝２，Ｘ（ｋ）β≡３（ｍｏｄ４）），这里，只是ＧＬ（ｅｐｉ）Ｒ（分别地，ＧＬ（２ｒｉ）Ｒ）的Ｓｙｌｏｗｐ－子群，Ｐ（ｊ））同构于Ｐ＝∪ｉ∈Ｉｐｉ，Ｉ是可数集．     

An upper bound for the representation dimension of group algebras with elementary abelian Sylow p-subgroups

Abstract

In this article, we present an upper bound on the representation dimension of the group algebra of a group with an elementary abelian Sylow p-subgroup. Specifically, if k is a field of characteristic p and G is a group with elementary abelian Sylow p-subgroup P, we prove that the representation dimension of kG is bounded above by the order of P. Key to proving this theorem is the separable equivalence between the two algebras and some nice properties of Mackey decomposition.     

Isomorphisms of Sylow p-subgroups of the restricted linear group II. Main theorem

We consider the Sylow p-subgroups, obtained by completion, of the restricted linear group of a countable-dimensional vector space of countable cardinality over a finite field of characteristic p. The geometric approach of O'Meara is used to describe the isomorphisms of linear groups that are not rich or even sufficiently rich in transvections. We prove that between two isomorphic Sylow p-subgroups there is an isomorphism of standard form induced by some locally inner automorphism of the restricted linear group.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 418–421, March, 1990.     

Normalizers of p-subgroups in finite groups

In this paper we give a computational strategy for constructing the normalizer of a p-subgroup in a finite group. Received: 31 January 2008     

Finite groups with complemented minimal p-subgroups

Abstract

For a given prime p, we investigate the finite groups in which every minimal p-subgroup is complemented.     

On quantitative structure of small Ree groups

The main aim of this article is to study quantitative structure of small Ree Groups ²G₂(q). Here, we prove that small Ree groups are uniquely determined by their orders and the set of the number of elements of the same order.