Sylow theory and the nilpotency class of left nilpotent skew braces

Abstract

Let $X$ be a finite left nilpotent skew brace and let $\pi$ be a set of primes. We show that every Hall $\pi$-subgroup of the multiplicative group $(X,\cdot )$ is a Hall $\pi$-subbrace of $X$. This extends \cite[Theorem 11]{CDDFT} by removing the solvability assumption. As an application, we obtain an upper bound for the left nilpotency class of $X$ in terms of the left nilpotency classes of its Sylow $p$-subbraces for every prime $p$ dividing the order of $X$. We also show that every $p$-subbrace of $X$ is contained in some Sylow $p$-subbrace.