Compact groups with many elements of bounded order

2020/01/17 by Malekan, Meisam Soleimani, Abdollahi, Alireza, Ebrahimi, Mahdi
#20E18 #20P05 #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2001.06508

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 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]). 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.

Related

• Profinite groups with many elements of bounded order
• Compact groups all elements of which are almost right Engel
• The Haar Measure Problem
• Almost Engel compact groups
• Nonmeasurable subgroups of compact groups