Book contents
- Frontmatter
- Contents
- Introduction
- Addresses of registered participants
- Addresses of non-participating authors
- Programme of lectures
- Conference photograph and key
- Symmetric presentations and orthogonal groups
- A constructive recognition algorithm for the special linear group
- Relations in M666
- A survey of symmetric generation of sporadic simple groups
- Harish-Chandra theory, q-Schur algebras, and decomposition matrices for finite classical groups
- The Meataxe as a tool in computational group theory
- Branching rules for modular projective representations of the symmetric groups
- Characters and surfaces: a survey
- On the characterization of finite groups by characters
- Finite linear groups of small degree
- Minimal parabolic systems for the symmetric and alternating groups
- Probabilistic methods in the generation of finite simple groups
- Condensing tensor product modules
- Intersections of Sylow subgroups in finite groups
- Anatomy of the Monster: I
- An integral ‘Meat-axe’
- Finite rational matrix groups: a survey
- Chamber graphs of sporadic group geometries
- An Atlas of sporadic group representations
- Presentations of reductive Fischer groups
- A brief history of the ATLAS
Intersections of Sylow subgroups in finite groups
Published online by Cambridge University Press: 19 May 2010
- Frontmatter
- Contents
- Introduction
- Addresses of registered participants
- Addresses of non-participating authors
- Programme of lectures
- Conference photograph and key
- Symmetric presentations and orthogonal groups
- A constructive recognition algorithm for the special linear group
- Relations in M666
- A survey of symmetric generation of sporadic simple groups
- Harish-Chandra theory, q-Schur algebras, and decomposition matrices for finite classical groups
- The Meataxe as a tool in computational group theory
- Branching rules for modular projective representations of the symmetric groups
- Characters and surfaces: a survey
- On the characterization of finite groups by characters
- Finite linear groups of small degree
- Minimal parabolic systems for the symmetric and alternating groups
- Probabilistic methods in the generation of finite simple groups
- Condensing tensor product modules
- Intersections of Sylow subgroups in finite groups
- Anatomy of the Monster: I
- An integral ‘Meat-axe’
- Finite rational matrix groups: a survey
- Chamber graphs of sporadic group geometries
- An Atlas of sporadic group representations
- Presentations of reductive Fischer groups
- A brief history of the ATLAS
Summary
Abstract
For every finite group G and every prime p, ip(G) ≤ 3, where ip(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, ip(G) ≤ 2.
Introduction
Let G be a finite group and p be a prime. Denote by Op(G) the intersection of all Sylow p-subgroups of G and by ip(G) the smallest number i such that Op(G) is equal to the intersection of i Sylow p-subgroups. Obviously, ip(G) = 1 if and only if G has an unique Sylow p-subgroup. J. Brodkey [2] proved that ip(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 ip(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 ip(G) = 2 for every prime p dividing the order of G.
Theorem 2 [9] For every finite group G and every prime p, ip(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.
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- The Atlas of Finite Groups - Ten Years On , pp. 191 - 197Publisher: Cambridge University PressPrint publication year: 1998