played by $(D(F)_d)^\wedge\times \biggl(D(t(G_0))_d\biggr)^\wedge$; $G_2$ is here identified with ${{\mathbb T\,}}^{r_0(G)}$ and $\pi$ is $\chi^{r_0(G)}$. Lemma \[enumer\] then provides a ${G_{\delta}}$-dense subgroup $\widetilde{H}$ of $\biggl(D(F)_d\biggr)^\wedge\times \biggl(D(t(G_0))_d\biggr)^\wedge$ such that both $\widetilde{H}$ and $\chi^{r_0(G)}(p_1(\widetilde{H}))$ have property ${{\mathbf \sharp}}$. This subgroup generates a pseudocompact topology $\mathcal{T}_{_{\widetilde{H}}}$ on $D(F)\oplus D(t(G_0))$ with property ${{\mathbf \sharp}}$ that makes $F$ pseudocompact (the induced topology on $F$ is just the topology inherited from $\chi^{r_0(G)}(p_1(\widetilde{H}))$). Since $G_0$ sits between $F$ and $D(F)\oplus D(t(G_0))$, it follows that the restriction of $\mathcal{T}_{_{\widetilde{H}}}$ to $G_0$ is pseudocompact and has property ${{\mathbf \sharp}}$.

By Proposition \[prop:boundedtorsion\] the bounded group ${\displaystyle \bigoplus_{\alpha(p_1^{n_1})}{{\mathbb Z}}(p_1^{n_1})\bigoplus
\cdots \bigoplus_{\alpha(p_k^{n_k})}{{\mathbb Z}}(p_k^{n_k})}$ also admits a pseudocompact group topology with property ${{\mathbf \sharp}}$ and the theorem follows.

Dikranjan and Shakmatov [@dikrshak05] prove under a set-theoretic axiom called $\nabla_\kappa$ (that implies ${\mathfrak{c}}=\omega_1$ and $2^{\mathfrak{c}}=\kappa$ with $\kappa$ being any cardinal $\kappa\geq \omega_2$) that every pseudocompact group of cardinality at most $2^{\mathfrak{c}}$ has a pseudocompact group topology with no infinite compact subsets. It follows from Theorem \[main\] that the result is true in ZFC, even for larger cardinalities.

Let $G$ be a pseudocompact Abelian group of cardinality $|G|\leq 2^{2^{\mathfrak{c}}}$. Then $G$ admits a pseudocompact topology with property ${{\mathbf \sharp}}$ (and thus a pseudocompact topology 