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Let $G$ be a finite group. It has been shown that:

  1. If the probability that two randomly selected elements of $G$ generate an abelian group is greater than $5/8$, $G$ is abelian.
  2. If the probability that two randomly selected elements of $G$ generate a solvable group is greater than $11/30$, $G$ is solvable.
  3. If the probability that two randomly selected elements of $G$ generate a nilpotent group is greater than $1/2$, $G$ is nilpotent.
  4. If the probability that two randomly selected elements of $G$ generate a group of odd order is greater than $11/30$, $G$ has odd order.

A proof of the first result, more often stated with "two randomly selected elements commute" in place of "generate an abelian group," can be found in this blog post by John Baez, who credits it to the paper "On some problems of a statistical group-theory, IV" by Erdős and Turán. The proofs of results 2-4 can be found in the paper The Probability of Generating a Finite Soluble Group by Guralnick and Wilson.

The proof of result 1 generalizes quite readily to the case where $G$ is a compact Hausdorff topological group (equipped with a Haar measure $\mu$). The details of such a proof can be found in the question Applications of the 5/8 Theorem by diracdeltafunk. In my view, this is due to the fact that the two main bounds used in the proof (that $[G \colon Z(G)] \geq 4$ for all nonabelian groups $G$ and $[G \colon C_{G}(g)] \geq 2$ for all noncentral elements $g \in G$) hold for all groups, and not just finite ones.

On the other hand, the proofs of results 2-4 do not seem to admit an easy generalization to the case of compact Hausdorff $G$: many propositions use the finiteness of $G$, and some of the related main results even use the Classification of Finite Simple Groups.

In view of these differences, my question is:

What are some other results in probabilistic group theory which are stated for finite groups, but have a proof that can easily be generalized to the case of compact Hausdorff groups, and what are some results with obvious obstacles to doing so?

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  • $\begingroup$ What would even be the analogue of 4 for infinite groups? $\endgroup$
    – Wojowu
    Jan 15, 2022 at 10:34
  • $\begingroup$ @Wojowu Since the conditions "having only elements of odd order" and "having odd order" are equivalent for finite groups, I'm thinking it might be something like: "If the probability that two elements of a compact Hausdorff group $G$ generate a subgroup of odd order is $11/30,$ then every element of $G$ has odd order." $\endgroup$
    – ckefa
    Jan 15, 2022 at 23:35
  • $\begingroup$ Maybe you should ask first: Are there counterexamples to 2 and 3 for compact Hausdorff groups? $\endgroup$
    – user44143
    Jan 19, 2022 at 20:42
  • $\begingroup$ I’m not sure that the analogue of 4 is interesting — is there a compact Hausdorff group where the elements of finite order have positive measure? $\endgroup$
    – user44143
    Jan 19, 2022 at 20:45
  • $\begingroup$ @MattF. Good idea, I will ask for counterexamples in a separate question if no answers are provided here. To address your concern about the analogue of 4: if we take a finite, discrete group $G$, and consider the product of infinitely many copies of $G$, don't we get an (infinite) compact Hausdorff group where every element has finite order? $\endgroup$
    – ckefa
    Jan 20, 2022 at 0:07

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