Prove that G\S generates G **_Question- Let $G$ be a group and $S < G$ (proper subgroup) then prove that $G$\ $S$ generates $G$._** **TRY-** I only have to prove that $S$ $\subset$ <$G$ \ $S$>. So to the contrary let $s \in S$ but not in <$G$ \ $S$> be an element , so ......... where do I go from here? I was also considering action of G or S on set of cosets of S in G but that was not fruitful either.

By assumption, there is some $g ∈ G\setminus S$. Then $g^{-1} ∈ G\setminus S$ as well.

And for any $s ∈ S$, you also have $sg ∈ G\setminus S$, so just write $s = sg·g^{-1} ∈ (G \setminus S) · (G\setminus S)$.