TITLE: The complement of a proper subgroup generates the whole group
QUESTION [5 upvotes]: $H$ is subgroup of $G$ with $H$ not equal $G$.
Be $S=G-H$. I am being asked to prove that $\langle S \rangle=G$.
Some tip to solve this? I think in $S_3$ is possible but I can´t prove.

REPLY [5 votes]: Hint. You have to use the hypothesis that $H\ne G$ and show that any element of $G$ can be written as a product of elements in $S$ or inverse thereof.
If $g\in S$, then of course $g\in\langle S\rangle$; if $g\in H$, pick an element $s\in S$; then $t=sg\notin H$ (why?); but $s^{-1}t=\dots$