The FSZ properties of sporadic simple groups

Abstract

We investigate a possible connection between the [Formula: see text] properties of a group and its Sylow subgroups. We show that the simple groups [Formula: see text] and [Formula: see text], as well as all sporadic simple groups with order divisible by [Formula: see text] are not [Formula: see text], and that neither are their Sylow 5-subgroups. The groups [Formula: see text] and [Formula: see text] were previously established as non-[Formula: see text] by Peter Schauenburg; we present alternative proofs. All other sporadic simple groups and their Sylow subgroups are shown to be [Formula: see text]. We conclude by considering all perfect groups available through GAP with order at most [Formula: see text], and show they are non-[Formula: see text] if and only if their Sylow 5-subgroups are non-[Formula: see text].