Analogues of Sylow's first theorem, Cauchy's theorem, and Hall's theorem for skew braces

Abstract

We establish an unconditional analogue of Sylow's first theorem for finite skew braces, and deduce an analogue of Cauchy's theorem. We also prove an analogue of the existence part of Hall's theorem for finite skew braces with soluble additive and multiplicative groups. We make some observations regarding the number of Sylow subskew braces of a skew brace in various cases. By applying these results we streamline the classification of skew braces of order $pq$, where $p,q$ are distinct prime numbers.

8 pages. To appear in Bulletin of the London Mathematical Society