subgroups of finite cyclic group Let $G=(g)$ be a finite cyclic group generated by $g$ with $|G|=n$, and let $d \in \mathbb{N}$ with $d|n$, then an unique subgroup $H$ of $G$ with $|H|=d$ exists.
Proof of existence: $\exists m \in \mathbb{N}: md=N$. Since $G=(g)$ we have $g^k\neq g^l\ \forall 0< k\neq l\leq N$ and $g^k=g^{k+nl}\ \forall k,l \in \mathbb{N}$.
Hence $g^{mk} \neq g^{ml}\ \forall 0< k\neq l \leq d$ and $g^{mk}=g^{m(k+nl)}\ \forall k,l \in \mathbb{N}$. Therefore $H=(g^m)=\{g^{m1},...,g^{md}=e\}$ is a (cyclic) subgroup of $G$ with $|H|=d$.
Is this ok?
Proof of uniqueness: Let $H_1,H_2$ be two subgroups with order $d$. As subgroups of a cyclic group $H_1,H_2$ are also cyclic. Set $k_1,k_2$ to be the smallest positive exponents so that $g^{k_1} \in H_1, g^{k_2} \in H_2$ respectively.
I do not know how I can continue from there to proof uniqueness..any help? Or is the approach wrong?
 A: Consider the surjective homomorphism
$$\varphi\colon\mathbb{Z}\to G,\qquad \varphi(k)=g^k$$
that has $\ker\varphi=n\mathbb{Z}$, where $g$ is a generator of $G$ and $|G|=n$.
If $H$ is a subgroup of $G$ with $|H|=d$, $d\mid n$, then we can compose $\varphi$ with the canonical projection to $G/H$, getting
$$
\pi\circ\varphi\colon\mathbb{Z}\to G/H
$$
that has kernel equal to $m\mathbb{Z}$, where $md=n$, because $m\mathbb{Z}$ is the only subgroup of $\mathbb{Z}$ having index $m$ and $|G/H|=m=n/d$. Therefore
$$
\varphi^{-1}(H)=m\mathbb{Z}
$$
so $H=\varphi(m\mathbb{Z})$, thus proving uniqueness.
A: Here is another solution to this old problem:
As every finite cyclic group of order $n$ is isomorphic to $\mathbb{Z}_n:=\mathbb{Z}/n\mathbb{Z}$ with the sum of integers modulo $n$, it is suffices to show this for $G=\mathbb{Z}_n$
Suppose $n=kp$, for $p,k\in\mathbb{N}$. If $p=1$ or $p=n$ then the corresponding subgroups of order $p$ are $(0)$ and $\mathbb{Z}_n$. for $1<p<n$ consider
$H_p:=(0,p,2p,\ldots,(k-1)p)$. This is clearly a subgroup for order $k$ for if
$$ap\equiv bp\mod(kp)$$
then $a\equiv b\mod k$ which means that the elements in $H_p$ do not repeat.
As for uniqueness, suppose $H'<\mathbb{Z}_n$ with $|H'|=k$. Let $p'$ be the smallest positive integer in $H'\cap\{1,1,\ldots,n-1\}$. Then $H'=\{0,p',2p',\ldots,(k-1)p'\}$ and $p'k=pk=n$. That is, $p=p'$.
