New classes of IYB groups

TL;DR

This paper proves new classes of finite groups are involutive Yang-Baxter (IYB) groups, expanding understanding of which groups admit structures of left braces, with implications for the open problem on solvable IYB groups.

Contribution

It establishes that certain finite solvable groups with specific Sylow subgroup properties are IYB groups, generalizing previous results and linking group structures to skew left braces.

Findings

Finite groups of odd order with certain Sylow subgroup properties are IYB.

Finite solvable groups with specific Sylow subgroup conditions are IYB.

Groups with the Sylow tower property are isomorphic to skew left brace multiplicative groups.

Abstract

It is proven that every finite group of odd order with all Sylow subgroups of nilpotency class at most two is an involutive Yang-Baxter group (IYB group for short), i.e. it admits a structure of left brace. It is also proven that every finite solvable group of even order with all Sylow subgroups of nilpotency class at most two and abelian Sylow 2-subgroups is an IYB group. These results contribute to the open problem asking which finite solvable groups are IYB, in particular they generalize a result of Ben David and Ginosar concerned with finite solvable groups with abelian Sylow subgroups. With the same techniques it is proven that every finite solvable group with all Sylow subgroups nilpotent of class at most two is isomorphic to the multiplicative group of a skew left brace of nilpotent type. It is also proven that every finite group with the Sylow tower property is isomorphic to the multiplicative group of a skew left brace of nilpotent type.