SUPERSOLUBLE AND RELATED COFINITARY GROUPS

Let D be a division ring (possibly commutative) and V an infinite-dime nsional left vector space over D. We consider irreducible subgroups G of GL(V) containing an element whose fixed-point set in V is non-zero but finite dimensional (over D). We then derive conclusions about cofi nitary groups, an element of GL(V) being cofinitary if its fixed-point set is finite dimensional and a subgroup G of GL(V) being cofinitary if all its non-identity elements are confinitary. In particular we sho w that an irreducible cofinitary subgroup G of GL(V) is usually imprim itive if G is supersoluble and is frequently imprimitive if G is hyper cyclic. The latter includes the case where G is hypercentral, which ap parently is also new.