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

Abstract

We show that any group $G$ with a non-$FS{Z}_m$ quotient by a central cyclic subgroup also provides a non-$FS{Z}_m$ 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 $FS{Z}_{p^j}$ group $F$ such that there is a central cyclic subgroup $A$ with $F/A$ not $FS{Z}_{p^j}$. We apply these results to regular wreath products to construct an $FSZ$ $p$-group which is not $FS{Z}^{+}$ for any prime $p>3$. These give the first known examples of $FSZ$ groups that are not $FS{Z}^{+}$. 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 $5^7$.

22 pages; Theorem 6.1 moved to 5.5, rest of section 6 removed due to an error