Prove that any group of order 15 is cyclic? I have multiple questions regarding: https://math.stackexchange.com/a/864985/997899
Q: Prove that any group of order 15 is cyclic.

Cauchy's theorem (group theory) : Let, $G $ is s a finite group and $p$ is a prime number dividing the order of $G$ (the number of elements in G), then $ G$ contains an element of order $p$.
$G$ be an abelian group . Let $a,b$  be elements of $ G$ with order m and n, respectively.
If $m$ and $n$are relatively prime, then show that the order of the element $ab$ is $ mn$.
Hence, $G$ contains an element of order $3$ say $a$ and an element of order $5$ say $b$. Then order of $ab$ is $3×5=15$
Since, $G$ is a group of order $15$ and contains an element $ab$ of order $15$.
Hence, $G=<ab $
Sylow's theorem is a another good options.
But an important corollary, if $order(G) =pq$ ,where$ p, q$ are primes with $p<q$  and $p $ doesn't divides $q-1$ , the the group $G$ is cyclic.
Here, $order(G)=3•5 $ and $3$ doesn't divide $( 5-1)=4$
