2020/12/27 by Aliréza Abdollahi, Abdollahi, Alireza, Meisam Soleimani Malekan +1
Computer Science · Mathematics · #20E18 #20P05 #43A05 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.2012.13886
openalex publication_date 2020/12/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Lévai and Pyber proposed the following as a conjecture: Let G be 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 (see Problem 14.53 of [The Kourovka Notebook, No. 19, 2019]). 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 [Arch. Math. (Basel) 75 (2000) 1-7] for n=2. Here we confirm the conjecture for n=3.