On left nilpotent skew braces of class 2

TL;DR

This paper develops a detailed structure theory for left nilpotent skew braces of class 2, establishing conditions under which they are centrally and right nilpotent, with precise bounds on their nilpotency classes.

Contribution

It introduces new structural results for left nilpotent skew braces of class 2, including bounds on their right nilpotency class and conditions for central nilpotency.

Findings

If of nilpotent type, then B is centrally nilpotent.

B is right nilpotent of class at most 2+mr.

If of abelian type, then B is right nilpotent of class 3.

Abstract

The main objective of this article is to initiate a detailed structure theory of left nilpotent skew braces $B$ of class $2$, i.e. skew braces with $B^3 = 0$. We prove that if $B$ is of nilpotent type, then $B$ is centrally nilpotent. In fact, we show that $B$ is right nilpotent of class at most $2+mr$, i.e. $B^{(2+mr+1)} = 0$, where $m$ and $r$ are the nilpotency classes of the additive group of $B$ and $B^2$, respectively. If $B$ is of abelian type, then $B$ is actually right nilpotent of class $3$, i.e. $B^{(4)} = 0$, and this bound is best possible.