1. ## subgroup generated

Let $S$ be a proper subgroup of $G$. If $G-S$ is the complement of $S$, prove that $=G$.

Let $S$ be a proper subgroup of $G$. If $G-S$ is the complement of $S$, prove that $=G$.
clearly $ \subseteq G.$ now let $g \in G.$ if $g \in G - S,$ then $g \in .$ if $g \in S,$ then choose $x \in G-S.$ then $x^{-1} \in G-S$ and $gx=y \in G-S.$ so: $g=yx^{-1} \in .$ Q.E.D.