The following problem is a general form of [another problem](https://mathoverflow.net/questions/209541/avoiding-countable-subgroups-of-a-group-homeomorphic-to-the-cantor-space) (motivation is available there). Initially, the problems were posted together, but the first one is solved below, a solution that does not apply to the more focused version of the problem, which is the needed scenario.


**Problem 1.** Let $G$ be an uncountable group, $H$ a countable subgroup of $G$, and $g\in G\smallsetminus H$.
Is there necessarily an element $x\in G\smallsetminus H$ such that $g\notin\langle H\cup \{x\}\rangle$?



Avoiding countable subgroups of general uncountable groups
**Counterexample for Problem 1:** According to [this answer](https://math.stackexchange.com/a/1226896/111012) Saharon Shelah constructed a "[Jónsson group](https://en.wikipedia.org/wiki/J%C3%B3nsson_cardinal)" of order $\aleph\_1,$ i.e., an uncountable group $G$ in which every proper subgroup is countable. Choose an element $g\in G\setminus\{e\}.$ Let $H$ be a maximal subgroup of $G$ not containing $g.$ Since $H$ is a proper subgroup of $G,$ $H$ is countable. By the maximality of $H,$ there is no element $x\in G\setminus H$ with $g\notin\langle H\cup\{x\}\rangle.$ 


