Is there a group $G$ and subgroups $H\_1, \dots, H\_n \leq G$ for some $n \in \mathbb{N}$, such that $[G : H\_i] = \infty$ for each $1 \leq i \leq n$, and $$G = \bigcup\_{i=1}^n H\_i \ \ ?$$ 



Can a group be a union of finitely many subgroups of infinite index?
It may well be a lot easier than this, but it follows from the answer to the weaker question [Can a group be a finite union of (left) cosets of infinite-index subgroups?](https://mathoverflow.net/questions/17396/can-a-group-be-a-finite-union-of-left-cosets-of-infinite-index-subgroups?rq=1) that it's not possible.


