Roots of unity in GL(2) modulo center I am aware of that fact that if $p>n+1$, then there are no elements of order $p$ in $GL_n(\mathbb{Z}_p)$, where $\mathbb{Z}_p$ is the ring of $p$-adic integers. I want to show that the same holds for $PGL_n(\mathbb{Z}_p)$ which is $GL_n(\mathbb{Z}_p)$ modulo its center. Any help will be appreciated.
Proof in the $GL_n$ case is as follows:
For $ X \in GL_n(\mathbb{Z}_p)$, suppose $X^p=1$. Then, $X^p-1=(X-1)(X^{p-1}+\cdots+1)$  and $(X^{p-1}+\cdots+1)$ is irreducible over $\mathbb{Z}_p$. So the minimal polynomial of $X$ is of degree $p-1$. But the characteristic polynomial of $X$ is of degree $n$ and hence $n \geq p-1$.
But things are getting complicated because of the center. Any help or reference?
 A: The argument essential works the same way. Let's use the fact that $\mathrm{GL}_n(\mathbb{Z}_p)\to \mathrm{PGL}_n(\mathbb{Z}_p)$ is surjective.
Suppose $X\in \mathrm{PGL}_n(\mathbb{Z}_p)$ satisfies $X^p=I_n$, and let $Y\in \mathrm{GL}_n(\mathbb{Z}_p)$ be a lift of $X$, which always exists by surjectivity. Then there exists $a\in \mathbb{Z}_p$ such that $Y^p=aI_n$ lies in the center. This implies that the minimal polynomial of $Y$ divides $t^p-a$. If $a$ is not a $p^{th}$ power, this polynomial is irreducible, and as in your case we obtain $n\geq p$.
If $a=b^p$ for some $b\in \mathbb{Z}^\times_p$, then $$t^p-b^p=(t-b)(t^{p-1}+bt^{p-2}+\cdots +b^{p-2}t+b^{p-1}).$$
If the minimal polynomial divides $t-b$, then $Y=bI_n$ so that $X=I$ in the quotient. Otherwise the minimal poly divides the second factor, which we claim is also irreducible. In particular, we get $n\geq p-1$ in this case. The upshot is that as soon as $p>n+1$, you get the result.
To see that the polynomial $g(t)=t^{p-1}+bt^{p-2}+\cdots +b^{p-2}t+b^{p-1}$ is irreducible over $\mathbb{Z}_p$, suppose it factors as $g(t) = p(t)q(t)$ with both factors having positive degree. Then $$ b^{p-1}(t^{p-1}+t^{p-2}+\cdots +t+ 1) = g(bt) = p(bt)q(bt)$$
gives a non-trivial factorization of $t^{p-1}+t^{p-2}+\cdots +t+ 1$ in $\mathbb{Z}_p[t]$, a contradiction.
