Cyclic Groups of Order p q/Proof 1

Lemma
By Sylow $p$-Subgroup is Unique iff Normal, $H$ and $K$ are normal subgroups of $G$.

Let $H = \gen x$ and $K = \gen y$.

To show $G$ is cyclic, it is sufficient to show that $x$ and $y$ commute, because then:
 * $\order {x y} = \order x \order y = p q$

where $\order x$ denotes the order of $x$ in $G$.

Since $H$ and $K$ are normal:


 * $x y x^{-1} y^{-1} = \paren {x y x^{-1} } y^{-1} \in K y^{-1} = K$

and


 * $x y x^{-1} y^{-1} = x \paren {y x ^{-1} y^{-1} } \in x H = H$

Now suppose $a \in H \cap K$.

Then:

where $e$ is the identity of $G$.

Thus:
 * $x y x^{-1}y^{-1} \in K \cap H = e$

Hence $x y = y x$ and the result follows.