Let $G$ be a $p$-group contained in $S\_{p^m}$ for some $m\in \mathbb{N}$, this is, the symmetric group of degree $p^m$. Assume that the lower central series, LCS for short, of $G$ is well understood.


Is there anything known about the LCS of $W(G):=G\wr C\_p$, where $C\_p$ is a cyclic group of order $p$, and $\wr$ is the (right regular) permutation action of $C\_p$ on $G\times \dotsb\times G$. That is, if $(g\_1,g\_2,\dotsc, g\_p)\in G\times \dotsb\times G$, then the action of $C\_p=\langle x\rangle$ on the base group $G\times \dots\times G$ is defined by
$(g\_1,g\_2,\dotsc, g\_p)^x=(g\_p,g\_1,\dotsc,g\_{p-1})$.


I am particularly interested in the indices $|\gamma\_i(W(G)):\gamma\_{i+1}(W(G))|$ for $i \in \mathbb{N}$.


I have been searching for some bibliography or papers on the topic but I have not been able to find anything.



Is there anything known about the lower central series of a group $G\wr C_p$?
Here is a solution in the special case in which each lower central factor $\gamma\_i(G) / \gamma\_{i+1}(G)$ is elementary abelian.


First consider the case in which $G$ is elementary abelian, written additively, so $G \cong \mathbf F\_p^d$ for some $d$. Let $x$ be a generator of $C\_p$. Then for $g \in G^p$ we have $[g,x] = \Delta(g)$, where $\Delta = \sigma-1$ and $\sigma : G^p \to G^p$ is the linear map defined by $\sigma(g\_1, \dots, g\_p) = (g\_p, g\_1, \dots, g\_{p-1})$. Therefore
$$\gamma\_i(G \wr C\_p) = \Delta^i(G^p) \qquad (i \ge 2).$$
By the $\mathbf{F}\_p[X]$ polynomial identity
$(X-1)^{p-1} = 1 + X + \cdots + X^{p-1}$
we have
$$\Delta^{p-1} = 1 + \sigma + \cdots + \sigma^{p-1}.$$
Therefore
$$\Delta^{p-1}(g) = (s, \dots, s), \qquad \text{where}~s=g\_1 + \cdots + g\_p.$$
Therefore $\gamma\_{p-1}(G \wr C\_p) = \Delta^{p-1}(G^p)$ is just the diagonal copy of $G$ in $G^p$, which is also the kernel of $\Delta$. It follows that the lower central factors of $G \wr C\_p$ are $G \times C\_p, G, \dots, G$.


Now consider the general case. Let $G = \Gamma\_1 \ge \cdots \ge \Gamma\_k = 1$ be the lower central series of $G$. Then $C\_p$ acts on each factor $\Gamma\_i^p / \Gamma\_{i+1}^p$, and the lower central series of $\Gamma\_i / \Gamma\_{i+1} \wr C\_p$ is as described above. By lifting to $G$ we get a long central series:
$$G \wr C\_p = \Gamma\_1^p C\_p \ge \Delta(\Gamma\_1^p) \Gamma\_2^p \ge \Delta^2(\Gamma\_1^p) \Gamma\_2^p \ge \cdots \ge \Delta^{p-1}(\Gamma\_1^p) \Gamma\_2^p \ge \Gamma\_2^p \ge \Delta(\Gamma\_2^p) \Gamma\_3^p \ge \cdots.$$
I claim this is in fact the lower central series. To see this it suffices to observe that $\Delta^i(\Gamma\_j^p)\Gamma\_{j+1}^p$ (for $i \le p-1$) contains the preimage $D$ in $\Gamma\_j^p$ of the diagonal subgroup of $\Gamma\_j^p/\Gamma\_{j+1}^p$, and $[D, G^p] = \Gamma\_{j+1}^p$.


In general if $\gamma\_i(G)/\gamma\_{i+1}(G)$ is not elementary abelian it seems more complicated. For example $C\_{p^2} \wr C\_p$ seems to have lower central factors $C\_{p^2} \times C\_p, C\_p, C\_p, \dots, C\_p$.


