if, $k=p^n$ for some $p\in {{\mathbb P}}$ and some positive integer $n$.

For a set $X$ and a cardinal number $\alpha$, $[X]^{\alpha}$ stands for the collection of all subsets of $X$ with cardinality $\alpha$.

Following Tkachenko [@tkac88], we say that a subgroup $H$ of a topological group $G$ is $h$-embedded if every homomorphism of $H$ to the unit circle ${{\mathbb T\,}}$ can be extended to a *continuous* homomorphism of $G$ to ${{\mathbb T\,}}$. If $G$ is totally bounded and $H$ is $h$-embedded in $G$, then the topology of $H$ must equal the maximal totally bounded topology of $H$ (or, using van Douwen’s terminology, $H=H^\sharp$).

The cardinal function $m(\alpha)$ will be often used. The cardinal $m(\alpha)$ is defined for every infinite cardinal $\alpha$ as the least cardinal number of a ${G_{\delta}}$-dense subset of a compact group $K_\alpha$ of weight $\alpha$. It is proved in [@comfrobe85] that this definition does not depend on the choice $K_\alpha$ and therefore makes sense. The same reference contains proofs of the following basic essential features of $m(\alpha)$: $$\log(\alpha)\leq m(\alpha)\leq (\log(\alpha))^\omega \quad \mbox{
and }\quad  \cf(m(\alpha))>\omega, \quad \mbox{ for every
}\alpha\geq \omega.$$ These inequalities have a much simpler form if *Singular Cardinal Hypothesis* (SCH) is assumed. SCH 