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I am looking for a reference for the maximal order of an element in PSL(2, $q$), where $q$ is prime power.

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Could you perhaps give a bit more background or motivation to this question? – Yemon Choi Feb 16 '13 at 17:55

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).

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I have edited my answer, giving you some hints where to look. However I don't have the books at hand, so I cannot check. – Stefan Kohl Feb 16 '13 at 11:24

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