Embeddings of finite groups in $B_n/{\Gamma }_k(P_n)$ for $k=2,3$

Abstract

Let $n\ge 3$. 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 ${\mathbb{Z}}_{p^r}{⋊}_{\theta }{\mathbb{Z}}_d$, where the action $\theta$ is injective, $p$ is an odd prime (resp. $p\ge 5$ 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 ${\mathbb{Z}}_9⋊{\mathbb{Z}}_3$, where the action is given by multiplication by $4$, and the Heisenberg group mod $3$.