The FSZ properties of sporadic simple groups

TL;DR

This paper explores the FSZ properties of sporadic simple groups, establishing which are FSZ or not, and linking these properties to their Sylow subgroups, with implications for small perfect groups.

Contribution

It provides new proofs for non-FSZ properties of certain sporadic groups and clarifies the relationship between FSZ properties and Sylow subgroups in simple and perfect groups.

Findings

G2(5) and S6(5) are not FSZ, as are their Sylow 5-subgroups.

All other sporadic simple groups are FSZ.

Non-FSZ property in small perfect groups correlates with their Sylow 5-subgroups.

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-$FSZ$.