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<p>Let ℭ be a class of finite groups which is closed for subgroups, quotients and direct products. Given a profinite group <i>G</i> and an element <i>x</i> ∈ <i>G</i>, we denote by <i>P</i><sub>ℭ</sub>(<i>x</i>, <i>G</i>) the probability that <i>x</i> and a randomly chosen element of <i>G</i> generate a pro-ℭ subgroup. We say that a profinite group <i>G</i> is ℭ-positive if <i>P</i><sub>ℭ</sub>(<i>x</i>, <i>G</i>) &gt; 0 for all <i>x</i> ∈ <i>G</i>. We establish several equivalent conditions for a profinite group to be ℭ-positive when ℭ is the class of finite soluble groups or of finite nilpotent groups. In particular, for the above classes, the profinite ℭ-positive groups are virtually prosoluble (resp., virtually nilpotent). We also draw some consequences on the prosoluble (resp. pronilpotent) graph of a profinite group.</p>

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Probabilistic properties of profinite groups

  • 期刊论文

  • 发表日期：

    2025年9月4日

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  • Eloisa Detomi,
  • Andrea Lucchini,
  • Marta Morigi,
  • Pavel Shumyatsky

摘要

Let ℭ be a class of finite groups which is closed for subgroups, quotients and direct products. Given a profinite group G and an element x ∈ G, we denote by Pℭ(x, G) the probability that x and a randomly chosen element of G generate a pro-ℭ subgroup. We say that a profinite group G is ℭ-positive if Pℭ(x, G) > 0 for all x ∈ G. We establish several equivalent conditions for a profinite group to be ℭ-positive when ℭ is the class of finite soluble groups or of finite nilpotent groups. In particular, for the above classes, the profinite ℭ-positive groups are virtually prosoluble (resp., virtually nilpotent). We also draw some consequences on the prosoluble (resp. pronilpotent) graph of a profinite group.