We show that any group $G$ with a non-$FSZ_m$ quotient by a central cyclicsubgroup also provides a non-$FSZ_m$ group of order $m|G|$ obtained as acentral product of $G$ with a cyclic group. We then construct, for every prime$p>3$ and $jinmathbb{N}$, an $FSZ_{p^j}$ group $F$ such that there is acentral cyclic subgroup $A$ with $F/A$ not $FSZ_{p^j}$. We apply these resultsto 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 thatare 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 appendixwe enumerate all non-$FSZ$ groups of order $5^7$.
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机译：我们表明，任何由中央周刊ubgroup的非费用群体的G $ Q $批准为$ M | $ M | $ M | $ M | $批准与循环集团的高额产品。然后，我们为每一个主要$ p> 3 $和$ j in mathbb {n} $，a $ fsz_ {p ^ j} $ group $ f $，使得有acentral循环子组$ a $ f $ f / a $ not $ fsz_ {p ^ j} $。我们应用这些结果级常规花圈产品来构建$ FSZ $ $ P $ -group，它不是$ FSZ ^ + $的任何PRIME $ P> 3 $。这些提供了$ FSZ $组的第一个已知的例子，而不是$ fsz ^ + $。我们还能够证明关于对称组的Sylow子组的$ FSZ $属性的一些部分结果。在附录我们中枚举所有非FSZ $ 5 ^ 7 $组。
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