Find the Existence and Uniqueness of a Cyclic subgroup

Question

Giving that $|H|$ is cyclic. If $|H|=n$ then for each $a>0$ such that $a|n$ there exist a unique subgroup of $H$ of order $a$. This subgroup is Cyclic subgroup $\langle x^d\rangle$ where $d= \frac{n}{a}$. I know in this proof that i have two show two things Existence and Uniqueness. Now to show existence of the subgroup i have to show this proposition $|x|=n$ and $a$ is a positive integer dividing $n$, then $|x^a|=\frac{n}{a}$. I have two question first how do i show this proposition and second how that help me show the existence of a subgroup.

Answer 1

By definition the order of an element is the smallest positive integer such that raising the element to that power is the identity. For $x^a$ it is true that $(x^a)^{n/a}=1$ by multiplicativeness of exponents, so we just need to know that this is the smallest power. But if there were a smaller one, say $m$, then $x^{ma}=1$ but $ma<n$, contradicting the fact that the order of $x$ is $n$.

The subgroup generated by $x^a$, which consists of all powers of this element, has $n/a$ elements, and we now know that this subgroup exists.