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

Abstract

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.

9 pages; we have added some new examples and the Sylow's Third Theorem; we have added an example showing that the number of Sylow sub-skew braces does not divide the order of the skew brace