Profinite Groups with Many Elements of Bounded Order*

Abstract

Lévai and Pyber [5] proposed the following as a conjecture (see also Problem 14.53 of [9]): if G is a profinite group such that the set of solutions of the equation xn = 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. We define a constant cn for all finite groups and prove that the latter conjecture is equivalent with a conjecture saying cn < 1. Using the latter equivalence we observe that correctness of Lévai and Pyber conjecture implies the existence of the universal upper bound 1/1-cn on the index of generalized Hughes-Thompson subgroup Hn of finite groups whenever it is non-trivial. It is known that the latter is widely open even for all primes n = p > 5. For odd n we also prove that Lévai and Pyber conjecture is equivalent to show that cn is less than 1 whenever cn is only computed on finite solvable groups. The validity of the conjecture has been proved in [5] for n = 2. Here we confirm the conjecture for n = 3. © 2022 AGTA.