Generation of iterated wreath products constructed from alternating, symmetric and cyclic groups
Jiaping Lu, Martyn Quick
TL;DR
We determine the minimal generating number $d(W)$ for the iterated wreath product $W = G_k\wr G_{k-1}\wr \dots\wr G_1$, where each $G_i$ is alternating, symmetric, or cyclic. Building on Lucchini’s framework for wreath products with abelian bases, the abelianization $A = (G_k/G_k')\times\cdots\times(G_2/G_2')$ is used to express explicit formulas: if $G_1 = A_4$, then $d(W)=\max(2, d(A), d_3(A)+1)$; if $G_1 = A_n$ with $n\ge5$, then $d(W)=\max(2, d(A))$; if $G_1 = S_n$ with $n\ge3$, then $d(W)=\max(2, d(A), d_2(A)+1)$; if $G_1$ is cyclic, then $d(W)=d(A)+1$. A corollary rewrites $d(W)$ purely in terms of $d(A)$ and $d_p(A)$ via $d(W/ W') = \max_p \{2, c_2+s, c_3+a_4, c_p\}$, where counts $c_p$, $a_4$, and $s$ track the corresponding factors among the $G_i$; this yields a uniform, combinatorial way to read the generating requirement from the group sequence. The results extend prior work on wreath products by unifying the abelian-base case with the non-abelian symmetric/alternating cases and provide a concrete, executable description of generation difficulty for these iterated constructions.
Abstract
Let $G_{1}$, $G_{2}$, ... be a sequence of groups each of which is either an alternating group, a symmetric group or a cyclic group and construct a sequence $(W_{i})$ of wreath products via $W_{1} = G_{1}$ and, for each $i \geq 1$, $W_{i+1} = G_{i+1} \operatorname{wr} G_{i}$ via the natural permutation action. We determine the minimum number $d(W_{i})$ of generators required for each wreath product in this sequence.