A Problem of W. R. Scott: Classify the Subgroup of Elements with Many Roots

TL;DR

This paper investigates the subgroup K(G) of an infinite group G, consisting of elements with almost all elements as roots, and explores conditions under which K(G) is non-trivial.

Contribution

It introduces the subgroup K(G) based on root properties and analyzes the criteria for its non-triviality in infinite groups.

Findings

K(G) contains elements with roots for almost all group elements.

Conditions for K(G) to be non-trivial are characterized.

The structure of K(G) varies depending on properties of G.

Abstract

Let G be an infinite group and let h and g be elements. We say that h is a root of g if some integer power of h is equal to g. We define K(G) to be the subgroup of all elements of G for which the number of elements which are not roots is of smaller cardinality than the cardinality of the group. That is, each element in K has almost every element in G as a root. This paper discusses the problem: When can K(G) be non-trivial?