Sylow Intersections in Finite Groups

Abstract

Ž . Here we study the meet semilattice generated by Syl G within the p subgroup lattice of the finite group G, where p is a fixed prime. We uncovered the results contained herein while investigating connections w x w x between Brodkey’s Theorem 2 and Ito’s paper 10 , both of which deal w x with intersections of Sylow subgroups. G. Robinson’s article 12 stimulated our search for deeper results. However, by the time our work became w x conclusive, scarcely any trace of the origins remained. The article 1 contains an earlier progress report, as well as more historical details. Our main result, Theorem 1.3 of this paper, implies that if p is an odd non-Mersenne prime, G is a finite p-solvable group, and P , P , P are 1 2 3 Sylow p-subgroups of G, then G possesses a Sylow p-subgroup P such 4 that P l P s P l P l P . 1 4 1 2 3 Our main result also considers when p s 2 or p is Mersenne. The same w x result cannot hold here as the examples in Ito’s paper 10 attest. However, we show that for such primes p in a p-solvable group, the intersection of any four Sylow p-subgroups can be obtained as the intersection of three Sylow p-subgroups with one Sylow p-subgroup in common. Our main result also considers conditions circumventing Ito’s examples so that the intersection of three Sylow subgroups can be obtained as the intersection of two. The final section gives examples to demonstrate how crucial p-solvability has been for our main result.