Nilpotent probability of compact groups
Alireza Abdollahi, Meisam Soleimani Malekan

TL;DR
This paper investigates the probability that randomly chosen elements in a compact group satisfy a nilpotency condition, establishing structural results about the group's open subgroups and connected components.
Contribution
It proves that positive nilpotent probability implies the existence of an open nilpotent subgroup of bounded class in profinite groups and characterizes the structure of connected components.
Findings
Positive nilpotent probability implies an open nilpotent subgroup in profinite groups.
The connected component of the group is abelian.
Existence of a closed normal nilpotent subgroup with specific properties.
Abstract
Let be any positive integer and a compact (Hausdorff) group. Let denote the probability that randomly chosen elements satisfy . We study the following problem: If then, does there exist an open nilpotent subgroup of class at most ? The answer is positive for profinite groups and we give a new proof. We also prove that the connected component of is abelian and there exists a closed normal nilpotent subgroup of class at most such that is open in .
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Taxonomy
TopicsFinite Group Theory Research · Advanced Topology and Set Theory · Limits and Structures in Graph Theory
