The Schur--Zassenhaus Theorem and Sylow's Third Theorem for Finite Skew Braces

In this short note we establish the Schur–Zassenhaus Theorem and Sylow’s Third Theorem for finite skew braces. More precisely, we prove that every Hall ideal of a finite skew brace admits a sub-skew brace complement, and more generally that every left ideal whose order is coprime to that of the Hall ideal can be embedded in such a complement. Using similar ideas we show that every left ideal of prime-power order is contained in a Sylow sub-skew brace. Finally, we prove that the number of Sylow $p$-sub-skew braces is congruent to $1$ modulo $p$, and provide examples showing that the corresponding containment property fails for arbitrary sub-skew braces.