Edited by R. T. Curtis
Edited by R. A. Wilson
Publisher: Cambridge University Press
Print Publication Year: 1998
Online Publication Date:May 2010
Online ISBN:9780511565830
Paperback ISBN:9780521575874
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511565830.019
Subjects: Algebra
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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.
pp. i-iv
pp. v-vi
pp. vii-viii
Addresses of registered participants : Read PDF
pp. ix-xiii
Addresses of non-participating authors : Read PDF
pp. xiii-xiii
Programme of lectures : Read PDF
pp. xiv-xiv
Conference photograph and key : Read PDF
pp. xv-xviii
Symmetric presentations and orthogonal groups : Read PDF
pp. 1-10
A constructive recognition algorithm for the special linear group : Read PDF
pp. 11-26
pp. 27-38
A survey of symmetric generation of sporadic simple groups : Read PDF
pp. 39-57
Harish-Chandra theory, q-Schur algebras, and decomposition matrices for finite classical groups : Read PDF
pp. 58-73
The Meataxe as a tool in computational group theory : Read PDF
pp. 74-81
Branching rules for modular projective representations of the symmetric groups : Read PDF
pp. 82-89
Characters and surfaces: a survey : Read PDF
pp. 90-118
On the characterization of finite groups by characters : Read PDF
pp. 119-138
Finite linear groups of small degree : Read PDF
pp. 139-148
Minimal parabolic systems for the symmetric and alternating groups : Read PDF
pp. 149-162
Probabilistic methods in the generation of finite simple groups : Read PDF
pp. 163-173
Condensing tensor product modules : Read PDF
pp. 174-190
Intersections of Sylow subgroups in finite groups : Read PDF
pp. 191-197
Anatomy of the Monster: I : Read PDF
pp. 198-214
An integral ‘Meat-axe’ : Read PDF
pp. 215-228
Finite rational matrix groups: a survey : Read PDF
pp. 229-248
Chamber graphs of sporadic group geometries : Read PDF
pp. 249-260
An Atlas of sporadic group representations : Read PDF
pp. 261-273
Presentations of reductive Fischer groups : Read PDF
pp. 274-287
A brief history of the ATLAS : Read PDF
pp. 288-293