$p$-elements in profinite groups

TL;DR

This paper studies properties of p-elements in profinite groups, showing conditions under which such groups contain open prosolvable or pro-p subgroups based on probabilities of element types.

Contribution

It establishes new criteria linking the probability of p-elements to the existence of open prosolvable or pro-p subgroups in profinite groups.

Findings

Positive probability of p-elements implies open prosolvable subgroup for odd p.

Existence of groups with high 2-element probability but not virtually prosolvable.

Certain probability conditions on element generation imply open pro-p subgroup presence.

Abstract

We investigate some properties of the $p$-elements of a profinite group $G$. We prove that if $p$ is odd and the probability that a randomly chosen element of $G$ is a $p$-element is positive, then $G$ contains an open prosolvable subgroup. On the contrary, there exist groups that are not virtually prosolvable but in which the probability that a randomly chosen element of $G$ is a 2-element is arbitrarily close to 1. We prove also that if a profinite group $G$ has the property that, for every $p$-element $x$, it is positive the probability that a randomly chosen element $y$ of $G$ generates with $x$ a pro-$p$ group, then $G$ contains an open pro-$p$ subgroup.