Suppose $G$ is a finite group and $\phi : G \rightarrow \mathbb{Z}_{10}$ is a surjective homomorphism. Then, $G$ normal subgroups of indexes $2$ and $5$.
Approach: Since $\phi$ is surjective then, by first isomoprhism thm, $G/\ker\phi \cong \mathbb{Z}_{10}$. Therefore, $|G| = 10|\ker\phi|$. Also, we know $10 \mid |G|$. Say $N \lhd G$. We need to show $[G:N] = 2$. Now, here I'm stuck, because I don't know how to show that $N$ has index $2$ in G? Can someone help me get to the next step?
Thanks!