Simple Skew Braces with Cyclic Sylow Subgroups

We study simplicity and splitting phenomena in finite skew braces under cyclicity assumptions on Sylow subgroups. We first classify finite simple skew braces whose multiplicative group is a $Z$-group. If the additive group is soluble, then the skew brace is either trivial of prime order or isomorphic to one of the two simple skew braces of order $12$ with additive group $A_4$ and multiplicative group $C_3\rtimes C_4$. If the additive group is insoluble, then it is necessarily isomorphic to $\operatorname{PSL}_2(p)$ for some prime $p\geq5$. This conclusion is sharp, since such examples exist for every prime $p\geq5$. We then consider the more general situation in which only a Sylow subgroup corresponding to the smallest prime divisor $p$ of the order is assumed to be cyclic. Under suitable hypotheses on the additive or multiplicative Sylow $p$-subgroup, we prove that the skew brace contains a Hall $p'$-ideal and splits as a semidirect product of this ideal with a Sylow $p$-subbrace. As a consequence, every finite simple skew brace satisfying one of these hypotheses is trivial of prime order. Moreover, as a consequence of our splitting theorem, we verify Byott's solvability conjecture for finite skew braces whose additive group has a cyclic Sylow $2$-subgroup.