# Some behaviors of FSZ groups under central products, central quotients, and regular wreath products

## TL;DR

This paper investigates the properties of FSZ groups under various group operations, constructs new examples of FSZ groups that are not FSZ+, and explores FSZ properties in symmetric groups and specific group orders.

## Contribution

It introduces new constructions of FSZ groups that are not FSZ+, and provides the first known examples of such groups, expanding understanding of FSZ group behaviors.

## Key findings

- Constructed FSZ groups with non-FSZ quotients by central cyclic subgroups.
- Provided examples of FSZ groups that are not FSZ+ for primes greater than 3.
- Enumerated all non-FSZ groups of order 5^7.

## Abstract

We show that any group $G$ with a non-$FSZ_m$ quotient by a central cyclic subgroup also provides a non-$FSZ_m$ group of order $m|G|$ obtained as a central product of $G$ with a cyclic group. We then construct, for every prime $p>3$ and $j\in\mathbb{N}$, an $FSZ_{p^j}$ group $F$ such that there is a central cyclic subgroup $A$ with $F/A$ not $FSZ_{p^j}$. We apply these results to regular wreath products to construct an $FSZ$ $p$-group which is not $FSZ^+$ for any prime $p>3$. These give the first known examples of $FSZ$ groups that are not $FSZ^+$. We are also able to prove a few partial results concerning the $FSZ$ properties for the Sylow subgroups of symmetric groups. In the appendix we enumerate all non-$FSZ$ groups of order $5^7$.