Abstract:We show that {every} subgroup of an $\Bbb R$-factorizable abelian $P$-group is topologically isomorphic to a {closed} subgroup of another $\Bbb R$-factorizable abelian $P$-group. This implies that closed subgroups of $\Bbb R$-factorizable $P$-groups are not necessarily $\Bbb R$-factorizable. We also prove that if a Hausdorff space $Y$ of countable pseudocharacter is a continuous image of a product $X=\DOTSB \prod@ \slimits@ _{i\in I}X_i$ of $P$-spaces and the space $X$ is pseudo-$\omega _1$-compact, then $nw(Y)\leq \aleph _0$. In particular, direct products of $\Bbb R$-factorizable $P$-groups are $\Bbb R$-factorizable and $\omega $-stable.
Keywords: $P$-space, $P$-group, pseudo-$\omega _1$-compact, $\omega $-stable, $\Bbb R$-factorizable, $\aleph _0$-bounded, pseudocharacter, cellularity, $\aleph _ 0$-box topology, $\sigma $-product
AMS Subject Classification: Primary 54H11, 22A05, 54G10; Secondary 54A25, 54C10, 54C25