CMA Research Report

MRR03-011

A note on Hall subgroups and subnormal products

Abstract:  If G is a finite group with Hall subgroup H and normal subgroups K and L, then simply by comparing orders we see that H ∩ KL = (H ∩ K)(H ∩ L). In this brief note we give a straightforward proof that the same result holds when K and L are subnormal, and consider how this might be generalised. As an immediate corollary of this observation and a theorem of Brewster, we obtain a new necessary and sufficient condition for a pair of subnormal subgroups to permute.

AMS Classification:  20D35
Date:  14 October 2003

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