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

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 ${Gamma_l(P_n)}_{lin mathbb{N}}$ 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 $3$.