# A family of non-FSZ finite symplectic groups

## TL;DR

This paper investigates specific finite symplectic groups over fields with certain prime characteristics, demonstrating they do not possess the FSZ property for particular parameters, thus expanding understanding of their algebraic structure.

## Contribution

It establishes that certain families of non-FSZ finite symplectic groups and their Sylow p-subgroups are non-FSZ for specified prime powers and parameters.

## Key findings

- Groups $ ext{Sp}_{p^j+1}(q)$ and $ ext{PSp}_{p^j+1}(q)$ are non-$FSZ_{p^j}$.
- Sylow p-subgroups of these groups are also non-$FSZ_{p^j}$.
- Results hold for odd primes $p
eq 1 mod 4$ and odd powers $q$ of $p$.

## Abstract

Let $p$ be an odd prime with $p\equiv1\bmod 4$. Then for any odd power $q$ of $p$ and a positive integer $j$ we show that the groups $\text{Sp}_{p^j+1}(q),\text{PSp}_{p^j+1}(q)$, and their Sylow $p$-subgroups are non-$FSZ_{p^j}$.