  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 $n=3$.
