Suppose that G is a group with $\displaystyle |G| = mp$ where $\displaystyle p$ is a prime and $\displaystyle 1 < m < p$. Prove that $\displaystyle G$ is not simple.
There are always Sylow p-subgroups. In this case, it follows immediately from Sylow's theorems that there is a unique one. Unique Sylow subgroups are normal.
But as you say, you haven't covered that yet, so I'm sure you're expected to make a more direct argument.
Since $\displaystyle p|mp$, there is a a subgroup $\displaystyle H$ of order $\displaystyle p$. A (sub)group of order $\displaystyle p$ is cyclic, thus $\displaystyle H$ is Abelian thus normal. Since we have a normal subgroup $\displaystyle H\ne\{e\}$ and $\displaystyle H\ne G$, $\displaystyle G$ is not simple.
Is this argument correct?
Let $\displaystyle H\leqslant G$ be such that $\displaystyle |H|=p$. It is trivial that there is a homomorphism $\displaystyle \phi:G\to \text{Sym}\left(G/H\right)$ by having $\displaystyle \phi_g(aH)=gaH$. Moreover, one can prove that $\displaystyle \ker\phi\subseteq H$. Now, since $\displaystyle p$ is prime and $\displaystyle \ker\phi\leqslant H$ we must have that $\displaystyle \ker\phi=\{e\}$ or $\displaystyle \ker\phi=H$. Suppose that $\displaystyle \ker\phi=\{e\}$ then $\displaystyle \text{im}(\phi)$ is a subgroup of $\displaystyle \text{Sym}\left(G/H\right)$ of order $\displaystyle mp$ and so $\displaystyle mp\mid m!$ but since $\displaystyle p$ is prime and $\displaystyle m<p$ this is impossible. Thus, $\displaystyle H=\ker\phi$ and so $\displaystyle \{e\}\triangleleft H\triangleleft G$ so that $\displaystyle G$ isn't simple.