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LÃ©vai 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 [V.ï¿½??D. Mazurov and E.ï¿½??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. LÃ©vai and L. Pyber, Profinite groups with many commuting pairs or involutions, Arch. Math. (Basel) 75 2000, 1ï¿½??7] for n=2. Here we study the conjecture for compact groups G which are not necessarily profinite and n=3; we show that in the latter case the group G contains an open normal 2-Engel subgroup.
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