Maximal subgroup growth of a few polycyclic groups

Abstract

We give here the exact maximal subgroup growth of two classes of polycyclic groups. Let Gk = ⟨x1, x2, . . . , xk | xixjxi⁻¹ for all i < j⟩, so Gk = ℤ⋊(ℤ⋊(ℤ⋊• • •⋊ℤ)). Then for all integers k ≥ 2, we calculate mn(Gk), the number of maximal subgroups of Gk of index n, exactly. Also, for infinitely many groups Hk of the form ℤ² ⋊ G₂, we calculate mn(Hk) exactly.