\alpha=|H_2|$, we can also enumerate a ${G_{\delta}}$-dense subgroup $D$ of $L$ (allowing repetitions if necessary) as $D=\{d_\eta \colon \eta <\alpha\}$. We now define the subgroup $\widetilde{H}$ of $G_1\times L$ as $$\widetilde{H}= \left\langle \,(f_\kappa + g_\eta,d_\eta)
 \colon \eta <\alpha, \kappa<\beta\,\right\rangle.$$ It is easy to check that $\widetilde{H}$ is a ${G_{\delta}}$-dense subgroup of $G_1\times L$ with $\widetilde{H}\cap \{0\}\times
 L=\{(0,0)\}$.

Since the homomorphism $p_1$ is continuous and establishes a group isomorphism between $\widetilde{H}$ and $H_1+H_2$, Lemma \[lem:quotients\] shows that $\widetilde{H}$ has property ${{\mathbf \sharp}}$. The same argument applies to the group $\pi\left(
p_1(\widetilde{H})\right)=\pi(H_1+H_2)$.

Let $\alpha\geq \omega$ be a cardinal. We say that *$\alpha$ satisfies property $(\ast)$ if:* $$\tag{*}\label{*} \mbox{there is a  cardinal $\kappa$ with }
\kappa^\omega \leq \alpha\leq 2^\kappa$$

Every cardinal $\alpha$ with $\alpha^\omega = \alpha$ satisfies property . This condition is equivalent to the condition $(m(\alpha))^\omega \leq \alpha$.

To apply Lemma \[enumer\] we need the following result:

\[comfgali\] Let $G=(G,\mathcal{T}_1)$ be a pseudocompact Abelian group with $w(G)=\alpha>\omega$, and set $$\sigma=\min\{r_0(N):N \text{ is a closed ${G_{\delta}}$-subgroup of $G$}\}.$$ If $\alpha^\omega\leq\sigma$ and if $\lambda\geq\omega$ satisfies $m(\lambda)\leq\sigma$, then $G$ admits a pseudocompact group topology $\mathcal{T}_2$ such that $w(G,\mathcal{T}_2)=\alpha+\lambda$ and $\mathcal{T}_1\bigvee\mathcal{T}_2$ is pseudocompact. Moreover, every closed ${G_{\delta}}$-subgroup of $(G,\mathcal{T}_1)$ is ${G_{\delta}}$-dense $(G,\mathcal{T}_2)$.

\[cor:comfgali\] Let $\sigma,\alpha$ and 