The FSZ properties of sporadic simple groups

Abstract

We investigate a possible connection between the $FSZ$ properties of a group and its Sylow subgroups. We show that the simple groups $G_2(5)$ and $S_6(5)$, as well as all sporadic simple groups with order divisible by $5^6$ are not $FSZ$, and that neither are their Sylow 5-subgroups. The groups $G_2(5)$ and $HN$ were previously established as non-$FSZ$ by Peter Schauenburg; we present alternative proofs. All other sporadic simple groups and their Sylow subgroups are shown to be $FSZ$. We conclude by considering all perfect groups available through GAP with order at most $10^6$, and show they are non-$FSZ$ if and only if their Sylow 5-subgroups are non-$FSZ$.

v2: 27 pages; minor title change; added the projective symplectic group PSp(6,5); various typo fixes; GAP code moved from an appendix into an earlier section; added in the code section that all groups of order less than 2016 (except possibly those of order 1024) have been verified as FSZ