Maximal subgroup growth of some metabelian groups

2018/07/09 by Andrew James Kelley, Kelley, Andrew James
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #math.GR

paper · pdf · doi:10.48550/arxiv.1807.03423

44 pages

arxiv created 2018/07/09 · arxiv updated 2018/07/11

Abstract

Let m_n(G) denote the number of maximal subgroups of G of index n. An upper bound is given for the degree of maximal subgroup growth of all polycyclic metabelian groups G (i.e., for \limsup (log m_n(G))/(log n), the degree of polynomial growth of m_n(G)). A condition is given for when this upper bound is attained. For G = ℤ^k \rtimes ℤ, where A ∈ GL(k,ℤ), it is shown that m_n(G) grows like a polynomial of degree equal to the number of blocks in the rational canonical form of A. The leading term of this polynomial is the number of distinct roots (in ℂ) of the characteristic polynomial of the smallest block.

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