Embeddings of finite groups in $B_n/\Gamma_k(P_n)$ for $k=2, 3$

TL;DR

This paper investigates conditions under which finite groups embed into specific quotients of braid groups, providing new embeddings for certain classes of groups and extending previous results in the field.

Contribution

It establishes new embedding results for finite groups into quotients of braid groups, especially for groups of odd order and certain semi-direct products, extending prior work.

Findings

Finite groups of odd order embed in B_{|G|}/Γ_2(P_{|G|})

Groups with order relatively prime with 6 embed in B_{|G|}/Γ_3(P_{|G|})

Explicit embeddings constructed for specific non-Abelian groups of order 27

Abstract

Let $n \geq 3$. In this paper, we study the problem of whether a given finite group $G$ embeds in a quotient of the form $B_n/\Gamma_k(P_n)$, where $B_n$ is the $n$-string Artin braid group, $k \in \{2, 3\}$, and $\{\Gamma_l(P_n)\}_{l\in \mathbb{N}}$ is the lower central series of the $n$-string pure braid group $P_n$. Previous results show that a necessary condition for such an embedding to exist is that $|G|$ is odd (resp. is relatively prime with $6$) if $k=2$ (resp. $k=3$), where $|G|$ denotes the order of $G$. We show that any finite group $G$ of odd order (resp. of order relatively prime with $6$) embeds in $B_{|G|}/\Gamma_2(P_{|G|})$ (resp. in $B_{|G|}/\Gamma_3(P_{|G|})$). The result in the case of $B_{|G|}/\Gamma_2(P_{|G|})$ has been proved independently by Beck and Marin. One may then ask whether $G$ embeds in a quotient of the form $B_n/\Gamma_k(P_n)$, where $n < |G|$ and $k \in \{2, 3\}$. If $G$ is of the form $\mathbb{Z}_{p^r} \rtimes_{\theta} \mathbb{Z}_d$, where the action $\theta$ is injective, $p$ is an odd prime (resp. $p \geq 5$ is prime) $d$ is odd (resp. $d$ is relatively prime with $6$) and divides $p-1$, we show that $G$ embeds in $B_{p^r}/\Gamma_2(P_{p^r})$ (resp. in $B_{p^r}/\Gamma_3(P_{p^r})$). In the case $k=2$, this extends a result of Marin concerning the embedding of the Frobenius groups in $B_n/\Gamma_2(P_n)$, and is a special case of another result of Beck and Marin. Finally, we construct an explicit embedding in $B_9/\Gamma_2(P_9)$ of the two non-Abelian groups of order $27$, namely the semi-direct product $\mathbb{Z}_9 \rtimes \mathbb{Z}_3$, where the action is given by multiplication by $4$, and the Heisenberg group mod $3$.