We study the following problem: For which Tychonoff spaces $X$ do the free topological group $F(X)$ and the free abelian topological group $A(X)$ admit a quotient homomorphism onto a separable and nontrivial (i.e., not finitely generated) group? The existence of the required quotient homomorphisms is established for several important classes of spaces $X$ , which include the class of pseudocompact spaces, the class of locally compact spaces, the class of $\unicode[STIX]{x1D70E}$ -compact spaces, the class of connected locally connected spaces, and some others.

We also show that there exists an infinite separable precompact topological abelian group $G$ such that every quotient of $G$ is either the one-point group or contains a dense non-separable subgroup and, hence, does not have a countable network.

我们研究以下问题：对于Tychonoff空间 $ X $来说 ，自由拓扑组 $ F（X）$ 和自由阿贝尔拓扑组 $ A（X）$ 承认在可分离且非平凡的商同态（即，不是有限的）生成）组？已为空间 $ X $的 几个重要类别建立了所需的商同态，其中包括伪紧缩空间的类别，局部紧致空间的类别， $ \ unicode [STIX] {x1D70E} $- 紧缩空间的类别，连通的局部连通空间的类别等。

我们还表明，存在着无限可分precompact拓扑阿贝尔群 $ G $ ，使得每一个商数 $ G $ 要么是一个点组或含有致密的不可分离的亚群，因此，不具有可数的网络。