1:Show that union of two subgroups may not be a group?
2:If G is a group and H is a subgroup of index 2 in G,prove that H is normal subgroup of G?
Take $\displaystyle S_3$ and $\displaystyle <(12)>$ with $\displaystyle <(13)>$.
Let $\displaystyle a\in G$. If $\displaystyle a\in H$ then $\displaystyle aH=Ha$. If $\displaystyle a\not \in H$ then $\displaystyle aH$ is the coset other than $\displaystyle H$, and $\displaystyle Ha$ is the coset other than $\displaystyle H$. Since there are only two cosets it means $\displaystyle aH=Ha$. Thus, $\displaystyle H$ is a normal subgroup of $\displaystyle G$.2:If G is a group and H is a subgroup of index 2 in G,prove that H is normal subgroup of G?