The Schur--Zassenhaus Theorem for finite skew braces

The aim of this short note is to prove an analogue of the existential part of the Schur--Zassenhaus Theorem for finite skew braces: we show that every Hall ideal of a finite skew brace admits a sub-skew brace complement.

As an application of similar ideas, we strengthen recent Sylow existence results by proving that every left ideal of prime-power order is contained in a Sylow sub-skew brace.