Intersections of Sylow subgroups in finite groups

Summary

Abstract

For every finite group G and every prime p, i_p(G) ≤ 3, where i_p(G) denotes the smallest number of Sylow p-subgroups of G whose intersection coincides with the intersection of all Sylow p-subgroups of G. For all simple groups (G, i_p(G) ≤ 2.

Introduction

Let G be a finite group and p be a prime. Denote by O_p(G) the intersection of all Sylow p-subgroups of G and by i_p(G) the smallest number i such that O_p(G) is equal to the intersection of i Sylow p-subgroups. Obviously, i_p(G) = 1 if and only if G has an unique Sylow p-subgroup. J. Brodkey [2] proved that i_p(G) ≤ 2 if a Sylow p-subgroup of G is abelian and N. Ito [5] found sufficient conditions for a finite solvable group to satisfy i_p(G) ≤ 2.

This paper discuss some recent results in this direction. The main theorems are the following:

Theorem 1 [10] If G is a simple non-abelian group then i_p(G) = 2 for every prime p dividing the order of G.

Theorem 2 [9] For every finite group G and every prime p, i_p(G) ≤ 3.

Section 2 contains a sketch of a proof of Theorem 1 which uses the Classification of Finite Simple Groups. Section 3 presents results about intersections of Sylow subgroups in arbitrary finite groups. We use the notation of the atlas [3].

Preliminary results

The following elementary results, the first of which is trivial, give a base for induction arguments in the proof of the main theorems of this paper.