On products of abelian skew braces

Abstract

The main objective of this paper is to study factorisations of skew left braces through abelian subbraces. We prove a skew brace theoretical analog of the classical Itô's theorem about product of two abelian groups: if $B=A_1{A}_2$ is a skew brace which is the product of two abelian skew subbraces $A_1$ and $A_2$, and $A_1$ is a left and right ideal of $B$, then the commutator ideal $[B,B{]}^B$ of $B$ is an abelian brace. If $A_1$ is a left (non-necessarily right) ideal of $B$, we show that there exists a strong left ideal of $B$ contained in $A_1$ or $A_2$. We also show factorisations of relevant ideals of factorised braces that are sums and products of abelian subbraces.

13 pages