Groups $\Gamma_n^4$: algebraic properties

TL;DR

This paper investigates the algebraic properties of groups  closely related to braid groups, establishing their nilpotency, finiteness, and torsion characteristics for sufficiently large n.

Contribution

It proves that for n,  are nilpotent finite 2-groups with 4-torsion and central derived subgroups, advancing understanding of their algebraic structure.

Findings

 are nilpotent finite 2-groups for n.

The groups have 4-torsion elements.

The derived subgroup ()' is central.

Abstract

In the paper, groups $\Gamma_n^4$ closely connected with braid groups are researched from algebraic point of view. More exactly, for $n\geqslant7$, it is proved that $\Gamma_n^4$ is a nilpotent finite $2$-group with $4$-torsion and that its subgroup $(\Gamma_n^4)'$ is central.