Abstract
We show that a group whose generalized torsion elements are torsion elements (which we call a -group) is a torsion-by- group, an extension of a 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.
Acknowledgments
The author would like to thank R. Bastos, C. Schneider, and D. Silveira for their stimulating works and discussions, and to R. Coulon and Y. Antolín for sharing their insight and answers to the Questions in the first version.
Disclosure statement
The author reports there are no competing interests to declare.
Notes
1 In this paper we adopt the convention that G is an extension of K by Q if there is an exact sequence .
2 We remark, however, that the normal subgroup K in Theorem 3 is not necessarily a generalized torsion group, because we require an element to be a generalized torsion element of G, but we do not require k to be a generalized torsion element of K.
3 Strictly speaking, is usually defined on [G, G]. We extend the domain of definition by defining if for some , and otherwise.