divisible weight in the structure of pseudocompact groups. One of its applications (Theorem 1.19 loc. cit.) is to prove that $r_0(G)$ is an admissible cardinal for every pseudocompact group $G$, a fact first proved by Dikranjan and Shakhmatov in [@dikrshak09].

Pseudocompact groups with property ${{\mathbf \sharp}}$
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The results of the previous sections will be used here to obtain sufficient conditions for the existence of pseudocompact group topologies with property ${{\mathbf \sharp}}$.

\[enumer\] Let $\pi \colon G_1\to G_2$ be a quotient homomorphism between two Abelian topological groups $G_1$ and $G_2$ and let $L$ be a compact Abelian group. Assume that the following conditions hold:

1.  $G_1$ contains a free ${G_{\delta}}$-dense subgroup $H_1$ such that $H_1$ and $\pi(H_1)$ are isomorphic and have property ${{\mathbf \sharp}}$.

2.  $G_1$ contains another free subgroup $H_2$ such that $H_1\cap H_2=\{0\}$, $H_1+H_2$ and $\pi(H_1+H_2)$ are isomorphic and have property ${{\mathbf \sharp}}$.

3.  $m(w(L))\leq |H_2|$.

Under these conditions the product $G_1\times L$ contains a ${G_{\delta}}$-dense subgroup $\widetilde{H}$ such that both $\widetilde{H}$ and $\pi\left(p_1(\widetilde{H})\right)$ have property ${{\mathbf \sharp}}$, where $p_1\colon G_1\times L\to G_1$ denotes the first projection.

We first enumerate the elements of $H_1$ and $H_2$ as $H_1=\{f_\beta\colon \kappa <\beta\}$ and $H_2=\{ g_\eta \colon \eta
<\alpha\}$. Since $m(w(L))\leq 