Supersoluble and Related Cofinitary Groups

Let D be a division ring (possibly commutative) and V an infinite‐dimensional 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 cofinitary 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 show that an irreducible cofinitary subgroup G of GL( V ) is usually imprimitive if G is supersoluble and is frequently imprimitive if G is hypercyclic. The latter includes the case where G is hypercentral, which apparently is also new.