with property ${{\sf S}_1}({{\mathcal{O}}_{nbd}},{\mathcal{O}})$. Then $G$ has the property ${{\sf S}_1}(\Omega,\Gamma)$.

This follows directly from Corollary \[rothbbdedgp\] part (1) that $(G,*)$ has property ${{\sf S}_1}({\mathcal{O}},{\mathcal{O}})$ since in the notation of that corollary $G=H$ and ${\mathcal{O}}_H={\mathcal{O}}$.

Recall that a ${\sf T}_0$ topological group is ${\sf T}_{3\frac{1}{2}}$. Since a compact ${\sf T}_3$-space is Rothberger if, and only if, it is scattered ([@BCM] Proposition 34), if, and only if, it is a $\gamma$-space (Theorem 4 of [@GN] and its Corollary), and since the countable union of compact $\gamma$-spaces is a $\gamma$-space (The union of two compact $\gamma$-spaces is a compact Rothberger space and thus a compact $\gamma$ space. Now apply Jordan’s theorem [@Jordan] Corollary 14), these topological groups are in fact $\sigma$-compact (thus $\sigma$-scattered) $\gamma$-groups. $\Box$

Thus for any cardinal number $\kappa$, the topological group ${{\mathbb R}}^{\kappa}$ contains $\sigma$-compact $\gamma$ subgroups of cardinality $\kappa$. It follows for example that the elements with finite support of any power of the integers is a $\sigma$-compact Rothberger bounded topological group.

\[rothbdedtorothbomega\] For each infinite cardinal number $\kappa$ there is a ${\sf T}_0$ topological group $(G,*)$ of cardinality $\kappa$ which is a $\sigma$-compact Rothberger space in all finite powers.

L. Babinkostova, Lj.D.R. Kočinac and M. Scheepers, *Combinatorics 