Dedicated to Bernhard Neumann on his 90th birthday.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 (g), PSL 2 (

A group is covered by a collection of subgroups if it is the union of the collection. The intersection of an irredundant cover of n subgroups is known to have index bounded by a function of n, though in general the precise bound is not known. Here we confirm a claim of Tompkinson that the correct bound is 16 when n is 5. The proof depends on determining all the 'minimal' groups with an irredundant cover of five maximal subgroups.

In this paper the concepts of control of transfer and fusion are introduced in the context of profinite groups. It is proved that a powerful Sylow p-subgroup controls transfer and that if the subgroup generated by the torsion elements of a Sylow p-subgroup is uniform~ then it controls fusion.

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