$\lambda$ be cardinals with $\alpha^\omega\leq \sigma$ and $m(\lambda)\leq \sigma$. If $H$ is a free, dense subgroup of ${{\mathbb T\,}}^\sigma$ with property ${{\mathbf \sharp}}$ and cardinality $\alpha$, then ${{\mathbb T\,}}^\sigma $ contains another subgroup $H_2$ with $H\cap H_2=\{0\}$, $|H_2|=\lambda+\alpha$ and such that $H+H_2$ has property ${{\mathbf \sharp}}$.

Let $F(\sigma)$ denote the free Abelian group of rank $\sigma$. We apply Theorem \[comfgali\] to the pseudocompact group $(F(\sigma),{\mathcal{T}_{\mbox{\tiny $H$}}})$ defined by $H$. We obtain thus a pseudocompact topology $\mathcal{T}_{_{H_2}}$ on $F(\sigma)$ induced by a subgroup $H_2$ of ${{\mathbb T\,}}^\sigma$ of cardinality $|H_2|=\alpha+\lambda$ such that ${\mathcal{T}_{\mbox{\tiny $H$}}}\bigvee \mathcal{T}_{_{H_2}}=\mathcal{T}_{_{H+H_2}}$ is pseudocompact. By Theorem \[hernmaca\] the subgroup $H+H_2$ has property ${{\mathbf \sharp}}$ and, since closed ${G_{\delta}}$-subgroups of ${\mathcal{T}_{\mbox{\tiny $H$}}}$ are ${G_{\delta}}$-dense in $\mathcal{T}_{_{H_2}}$, we also have that $H\cap
H_2=\{0\}$.

\[main\] Let $G$ be a pseudocompact Abelian group with dominant ranks $\gamma(p_1^{n_1}),\ldots, \gamma(p_k^{n_k})$ and suppose that $\gamma(p_i^{n_i})$, $1\leq i\leq k$, satisfy property . If $r_0(G)$ also satisfies property for some $\kappa$ with $m(|G_0|)\leq 2^\kappa$, then $G$ admits a pseudocompact topology with property ${{\mathbf \sharp}}$.

Decompose, following Theorem \[estrpseu\], $G$ as a direct sum $$G=\left(\bigoplus_{\gamma(p_1^{n_1})}{{\mathbb Z}}(p_1^{n_1})\bigoplus \cdots
\bigoplus_{\gamma(p_k^{n_k})}{{\mathbb Z}}(p_k^{n_k})\right) \bigoplus G_0$$ Let $F$ denote a free Abelian group of cardinality $r_0(G)$ contained in $G_0$ and denote by $D(F)$ 