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

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?