A Generalized Hughes Property of Finite Groups

Abstract

A Hughes cover for exponent p(pa prime number) of a finite group is a union of subgroups whose (non-empty) complement consists of elements of order p. A proper Hughes subgroup is an instance of a Hughes cover; and its parent group is soluble by a well-known result of Hughes and Thompson. More generally an earlier result of the authors shows that a group with a Hughes cover of fewer than psubgroups is soluble. This article treats the insoluble groups having a Hughes cover for exponent pwith exactly psubgroups: the almost simple groups with this property form a restricted class of projective special linear groups.

Key Words:

• Finite groups
• Hughes subgroup

Acknowledgments

The first named author thanks the Dipartimento di Matematica of the Università degli Studi di Firenze for its hospitality, and C.N.R. for financial support, while some of this work was done. We thank two referees for their helpful comments. Following the death of Prof. Fedri the first and third named authors have made minor revisions to the original version of this article.