Compact groups with many elements of bounded order

Abstract

Abstract Lévai 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 {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 [V. D. Mazurov and E. I. Khukhro, Unsolved Problems in Group Theory. The Kourovka Notebook. No. 19, Russian Academy of Sciences, Novosibirisk, 2019; Problem 14.53]). The validity of the conjecture has been proved in [L. Lévai and L. Pyber, Profinite groups with many commuting pairs or involutions, Arch. Math. (Basel) 75 2000, 1–7] for n = 2 {n=2} . Here we study the conjecture for compact groups G which are not necessarily profinite and n = 3 {n=3} ; we show that in the latter case the group G contains an open normal 2-Engel subgroup.