for $X \leq \text{SL}_{2}(\mathbb{Z}/p^{e}\mathbb{Z}) $ $n=2$ and $p \geq 5$ then $X=\text{SL}_{2}(\mathbb{Z}/p^{e}\mathbb{Z)}$

Question

Define $\theta$ s.t

$$ \theta :\mathbb{Z}/p^{e}\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z} $$

$$a+p^{e}\mathbb{Z} \to a + p\mathbb{Z}$$

We extend this to a h.m where:

$$\tilde{\theta} : \text{GL}_{n}(\mathbb{Z}/p^{e}\mathbb{Z}) \to \text{GL}_{n}(\mathbb{Z}/p\mathbb{Z}) $$ $$a_{ij} \to a_{ij}\theta$$

Now we assume $$X \leq \text{SL}_{n}(\mathbb{Z}/p^{e}\mathbb{Z})$$ where $$X\tilde{\theta} = \text{SL}_{n}(\mathbb{Z}/p\mathbb{Z})$$

then prove for $n=2$ and $p \geq 5$, that

$$X = \text{SL}_{2}(\mathbb{Z}/p^{e}\mathbb{Z})$$