G is a commutative group of order 2*N, how to prove G has a N-order quotient group?

Question

$G$ is a commutative group of order $2N$, how to prove $G$ has a $N$-order quotient group?

Now I know $G$ must has two subgroup, one is order $N$ and another is order $2$; now what should do to prove it has a $N$-order quotient group?

Answer 1

By Cauchy's theorem $G$ has a subgroup $H$ of order $2$. Now by Lagrange's theorem we have

$$[G:H]\cdot |H|=|G|$$

meaning $[G:H]=N$. Finally since $G$ is commutative then $H$ is normal and so $G/H$ is a quotient group of order $|G/H|=[G:H]$