Subgroup coverings of some linear groups

Abstract

A cover for a group is a collection of proper subgroups whose union is the whole group. A cover is minimal if no other cover contains fewer members. We term minimised a minimal cover with the property that substituting for a member of the cover by a proper subgroup of that member produces a collection which is no longer a cover. We here describe the minimised covers for the groups GL 2 ( q ), SL 2 ( q ), PSL 2 ( q ) and PGL 2 ( q ).