Maximal subgroup growth of a few polycyclic groups

Andrew James Kelley; Elizabeth Ciorsdan Dwyer Wolfe

paper

Maximal subgroup growth of a few polycyclic groups

arXiv:1911.07066

Abstract

We give here the exact maximal subgroup growth of two classes of polycyclic groups. Let $G_{k} = ⟨ x_{1}, x_{2}, ..., x_{k} ∣ x_{i} x_{j} x_{i}^{- 1} x_{j} for all i < j ⟩$ . So $G_{k} = Z ⋊ (Z ⋊ (Z ⋊ ... ⋊ Z)$ . Then for all $k \geq 2$ , we calculate $m_{n} (G_{k})$ , the number of maximal subgroups of $G_{k}$ of index $n$ , exactly. Also, for infinitely many groups $H_{k}$ of the form $Z^{2} ⋊ G_{2}$ , we calculate $m_{n} (H_{k})$ exactly.