On a group whose generalized torsion elements are torsion elements

Abstract

We show that a group whose generalized torsion elements are torsion elements (which we call a $TR^{*}$-group) is torsion-by-$R^{*}$ group, an extension of torsion group by a group without generalized torsion elements. We also discuss a generalized torsion group, a group all of whose non-trivial elements are generalized torsion elements.