Let $G=GL(n,q)$ be a generalized linear group whose center is a cyclic group of prime order $p$. Does there always exists an element $x\in G$ such that $C\_G(x)$ is a group of exponent $p$?



Centralizers of p-elements in general linear groups
As I mentioned in comments, either $q=3$ or $q$ is even. 


**Suppose that $q>n$**. Then $G$ contains a regular semisimple element $x$ with eigenvalues in $\mathbb{F}\_q$, then its centralizer is a maximal split torus of $G$, which is elementary abelian of order $(q-1)^n$. Thus the answer is **YES** in this case.


**Suppose that $q\leq n$**. Let's try to find an element $x$ such that $C\_G(x)$ has exponent $p$. Clearly $x$ must be semisimple. If $x$ has eigenvalues outside $\mathbb{F}\_q$, then it lies in a torus which contains a cyclic subgroup of order $q^i-1$ where $i>1$. This subgroup lies in $C\_G(x)$ which is a contradiction. Thus all eigenvalues of $x$ lie in $\mathbb{F}\_q$. Since $q\leq n$ we know that at least one of these eigenvalues is repeated. Now one can check that $|C\_G(x)|$ is divisible by $r$, the field characteristic, so again we have a contradiction. Thus the answer is **NO** in this case.


