Compact groups with many elements of bounded order

Meisam Soleimani Malekan; Alireza Abdollahi; Mahdi Ebrahimi

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arXiv · 2001.06508

Compact groups with many elements of bounded order

Meisam Soleimani Malekan·Alireza Abdollahi·Mahdi Ebrahimi

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Abstract

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]). The validity of the conjecture has been proved in [Arch. Math. (Basel) 75 (2000) 1-7] 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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Meisam Soleimani Malekan, Alireza Abdollahi, Mahdi Ebrahimi. 2020-01-17. Compact groups with many elements of bounded order. https://arxiv.org/abs/2001.06508

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