Maximal subgroup growth of a few polycyclic groups

2019/11/16 by Andrew James Kelley, Kelley, Andrew James, Elizabeth Ciorsdan Dwyer Wolfe +1
Computer Science · Mathematics · #Advanced Graph Theory Research #FOS: Mathematics #Group Theory (math.GR) #Mathematical Dynamics and Fractals #math.GR

paper · pdf · doi:10.48550/arxiv.1911.07066

arxiv created 2019/11/16 · openalex publication_date 2019/11/16 · arxiv updated 2019/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give here the exact maximal subgroup growth of two classes of polycyclic groups. Let G_k = ⟨ x₁, x₂, ..., x_k | x_ix_jx_i^-1x_j for all i < j ⟩. So G_k = ℤ \rtimes (ℤ \rtimes (ℤ \rtimes ... \rtimes ℤ). Then for all k ≥ 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 ℤ² \rtimes G₂, we calculate m_n(H_k) exactly.

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