Braces and symmetric groups with special conditions

TL;DR

This paper investigates the properties of symmetric groups and braces under specific identities like Raut and lri, establishing their impact on multipermutation levels and constructing examples with particular conditions.

Contribution

It characterizes the relationship between identities like lri and Raut in symmetric groups and braces, and constructs examples illustrating these properties.

Findings

Symmetric group G has multipermutation level 2 iff it satisfies lri.

Two-sided braces with Raut can be constructed without lri.

Finitely generated two-sided braces with lri have bounded multipermutation level.

Abstract

We study symmetric groups and left braces satisfying special conditions, or identities. We are particularly interested in the impact of conditions like $\textbf{Raut}$ and $\textbf{lri}$ on the properties of the symmetric group and its associated brace. We show that the symmetric group $G=G(X,r)$ associated to a nontrivial solution $(X,r)$ has multipermutation level $2$ if and only if $G$ satisfies $\textbf{lri}$. In the special case of a two-sided brace we express each of the conditions $\textbf{lri}$ and $\textbf{Raut}$ as identities on the associated radical ring $G_*$. We apply these to construct examples of two-sided braces satisfying some prescribed conditions. In particular we construct a finite two-sided brace with condition $\textbf{Raut}$ which does not satisfy $\textbf{lri}$. (It is known that condition $\textbf{lri}$ implies $\textbf{Raut}$). We show that a finitely generated two-sided brace which satisfies \textbf{lri} has a finite multipermutation level which is bounded by the number of its generators.