Let $G$ be a finite abelian group and $\widehat{G}$ the character group.
Let $S \subset \widehat{G}$ be a Galois-stable subset i.e. if $\chi \in S$, then the Galois conjugates $\chi^{\sigma} \in S$ for any $\sigma \in Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. Let $H\_{S} \subset \widehat{G}$ be the subgroup generated by $S$. 


We now consider a sequence $(G\_{i},S\_{i})$ as above with $|G\_{i}|\rightarrow \infty$. Suppose that there exists $\epsilon$ with $ 0 < \epsilon < 1$ such that $|H\_{S\_{i}}| \gg |G\_{i}|^{\epsilon}$. Then, is it necessary to have 
$|S\_{i}| \gg \ln(|G\_{i}|)$? Can we say something about optimal lower bound?



Characters and Galois stability
Suppose that $G\_i=C\_2^i$, and let $S\_i$ be a minimal generating set of $\widehat{G\_i}$. Since all characters are rational, $S\_i$ is stable, but we have $|H\_{S\_i}|=|G\_i|=2^i$, $|S\_i|=i\asymp\log|G\_i|$. So the bound $\log|G\_i|$ is in fact optimal.


