The question is:
Let $G$ be a group and N $\unlhd G$ with $N$ not contained in $Z(G)$.
Prove that:
a) if $N \cong \bf{C}_3$, then $G$ has a subgroup of index $2$.
b) if $N \cong \mathbb{Z}$, then $G$ has a subgroup of index $2$.
How would I go about proving this? Would I have to explicitely construct the subgroups, or is there some theorem or indirect existence argument I can use?
Edit: Do I somehow have to construct two cosets in G?