A generalised Hughes property for finite groups II

Research Report MRR96-065

A generalised Hughes property for finite groups II

R. A. Bryce, V. Fedri and L. Serena

Abstract: Let p be a prime and G a finite group properly containing a union $\cal A$ of r subgroups in such a way that the complement of $\cal A$ in G contains only elements of order p. Such a union in G we call a Hughes cover for exponent p. $\cal A$ generalises the notion of Hughes subgroup. When p>r earlier work (A Hughes-like property for finite groups, Proc. Edinburgh Math. Soc. 38 (1995), 533-541) shows that G has nilpotent normal p-complement, though for $p\le r$ insoluble G may exist. More recently (A generalised Hughes property for finite groups, Research Report 26/1996, Dipartimento di Matematica, Università degli Studi di Firenze) we have shown that when p=r and G is insoluble, G/F(G) is an almost simple group, also with a Hughes cover for exponent p, in which the socle is a projective special linear group of a rather restricted type. In this continuation of that work we show that these projective special linear groups are precisely all the almost simple groups which arise, and we examine the types of Hughes covers they may have.

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