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The OP is interested rather in a reference.
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Stefan Kohl ♦
Stefan Kohl ♦
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The keyword here is Singer cycle. -- A Singer cycle in ${\rm GL}(n,q)$ is an elementstandard reference for this type of order $q^n-1$. This results is the largest possible order of an element of

Bertram Huppert: ${\rm GL}(n,q)$Endliche Gruppen I, Band 134 der Grundlehren der mathematischen Wissenschaften, 1967, Springer-Verlag.

However I don't have that book at hand, so I cannot check. AsAnother source you are looking for element orders inmight wish to check is

Jean Dieudonne: ${\rm PSL}(n,q)$La Geometrie des Groupes Classiques, you need to divide that order by the product of $q-1$ and the size of the centre of ${\rm GL}(n,q)$ Ergebnisse der Mathematik und ihrer Grenzgebiete 5(i.e. $\gcd(n,q-1)$1963) to obtain the value you are looking for.

The keyword here is Singer cycle. -- A Singer cycle in ${\rm GL}(n,q)$ is an element of order $q^n-1$. This is the largest possible order of an element of ${\rm GL}(n,q)$. As you are looking for element orders in ${\rm PSL}(n,q)$, you need to divide that order by the product of $q-1$ and the size of the centre of ${\rm GL}(n,q)$ (i.e. $\gcd(n,q-1)$) to obtain the value you are looking for.

A standard reference for this type of results is

Bertram Huppert: Endliche Gruppen I, Band 134 der Grundlehren der mathematischen Wissenschaften, 1967, Springer-Verlag.

However I don't have that book at hand, so I cannot check. Another source you might wish to check is

Jean Dieudonne: La Geometrie des Groupes Classiques, Ergebnisse der Mathematik und ihrer Grenzgebiete 5(1963).

Source Link
Stefan Kohl ♦
Stefan Kohl ♦
  • 19.6k
  • 21
  • 75
  • 137

The keyword here is Singer cycle. -- A Singer cycle in ${\rm GL}(n,q)$ is an element of order $q^n-1$. This is the largest possible order of an element of ${\rm GL}(n,q)$. As you are looking for element orders in ${\rm PSL}(n,q)$, you need to divide that order by the product of $q-1$ and the size of the centre of ${\rm GL}(n,q)$ (i.e. $\gcd(n,q-1)$) to obtain the value you are looking for.