On Sylow intersections in finite groups
Abstract: In general, for a given prime and finite group , there need not be Sylow -subgroups and of with . In this paper we show that if is -soluble, and is not 2 or a Mersenne prime, then such Sylow -subgroups exist (also we give conditions guaranteeing the existence of such Sylow subgroups when is 2 or a Mersenne prime). We also show that if is not -soluble, but is odd and the components of are in a certain class of quasi-simple groups, then there are Sylow -subgroups and of with , unless perhaps is a Mersenne prime. When is -soluble, our work extends results of N. Itô .
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