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 $T{R}^{*}$-group) is a torsion-by-$R^{*}$ 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.

KEYWORDS:

• Generalized torsion
• non-trivial elements
• torsion elements
• torsion-by-$R^{\ast }$
• $T{R}^{\ast }$-group

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20F60
• 06F15
• 20F24

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 $1\to K\to G\to Q\to 1$.

2 We remark, however, that the normal subgroup K in Theorem 3 is not necessarily a generalized torsion group, because we require an element $k\in K\subset G$ 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, ${\text{scl}}_G$ is usually defined on [G, G]. We extend the domain of definition by defining ${\text{scl}}_G(g)=\frac{\text{scl}(g^k)}{k}$ if $g^k\in [G,G]$ for some $k>0$, and ${\text{scl}}_G(g)=\infty$ otherwise.

Additional information

Funding

The author is partially supported by JSPS KAKENHI Grant Numbers 19K03490, 21H04428, and 23K03110.