Profinite groups with many elements of bounded order

TL;DR

This paper investigates a conjecture relating the measure of solutions to $x^n=1$ in profinite groups to the existence of specific open subgroups, proving the conjecture for $n=3$ and exploring its implications for finite groups.

Contribution

It proves the conjecture for the case $n=3$, establishes equivalences involving a constant $c_n$, and discusses bounds on subgroup indices in finite groups.

Findings

Confirmed the conjecture for $n=3$.

Established the equivalence of the conjecture with bounds on $c_n$.

Linked the conjecture to the structure of finite groups and subgroup indices.

Abstract

L\'evai and Pyber proposed the following as a conjecture: Let $G$ be a profinite group such that the set of solutions of the equation $x^n=1$ has positive Haar measure. Then $G$ has an open subgroup $H$ and an element $t$ such that all elements of the coset $tH$ have order dividing $n$ (see Problem 14.53 of [The Kourovka Notebook, No. 19, 2019]). \\ We define a constant $c_n$ for all finite groups and prove that the latter conjecture is equivalent with a conjecture saying $c_n<1$. Using the latter equivalence we observe that correctness of L\'evai and Pyber conjecture implies the existence of the universal upper bound $\frac{1}{1-c_n}$ on the index of generalized Hughes-Thompson subgroup $H_n$ of finite groups whenever it is non-trivial. It is known that the latter is widely open even for all primes $n=p\geq 5$. For odd $n$ we also prove that L\'evai and Pyber conjecture is equivalent to show that $c_n$ is less than $1$ whenever $c_n$ is only computed on finite solvable groups. \\ The validity of the conjecture has been proved in [Arch. Math. (Basel) 75 (2000) 1-7] for $n=2$. Here we confirm the conjecture for $n=3$.