Take the tour ×  I am looking for a formula for the maximal
order of an element in GL(n,p), where p is prime.  I recall
seeing such a formula in a paper from the mid- or early 20th
century, but could not find again this reference. I will be
grateful for any hint.  share|improve this question add comment
1 Answer  Well, by Hamilton--Cayley, each matrix $A\in {\rm
GL}(n,p)$ generates an at most $n$-dimensional subalgebra
${\mathbb F}_p[A]\subseteq M(n,p)$ thus containing at most
$p^n-1$ nonzero elements. Hence the order of $A$ cannot exceed
$p^n-1$.  On the other hand, consider a degree $n$ monic
polynomial $P_n$ whose root is a generator $\xi$ of ${\mathbb
F}_{p^n}^*$. Then a matrix with $P_n$ as its characteristic
polynomial has order at least $p^n-1$ since $\xi$ is its
eigenvalue.  ADDENDUM. if you wish the order to be the power of
$p$, then the answer is $d=p^{\lceil \log_p n\rceil}$. Since the
order of $A$ is divisible by the multiplicative orders of its
eigenvalues, all the eigenvalues should be $1$. Hen