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The maximal order of an element of $\mathrm{GL}(n,\mathbb{F}_q)$ is $q^n-1$, where the characteristic of $\mathbb{F}_q$ is odd $p$. See here for a nice proof that uses the Cayley-Hamilton Theorem.

However, for $\mathrm{SL}(2,\mathbb{F}_q)$ the maximal order is $2p$ if $q=p$ and $q+1$ otherwise.

To see this, note that over $\mathbb{F}_{q^2}$ any such matrix $A$ is conjugate to an upper-triangular matrix. If it is not diagonal then its eigenvalues are repeated and $\pm 1$; thus its order is bounded by $2p$. If it is diagonal then the eigenvalues are in $\mathbb{F}_q$ or properly in the degree two extension. If the former then the order divides $|\mathbb{F}_q^*|=q-1$, and if the latter then $x\mapsto x^q$ generates the Galois group of the extension showing the eigenvalue $a$ satisfies $a^q=a^{-1}\implies a^{q+1}=1 \implies |a|{\large \mid} q+1$.

I did some searching on MO and other places online but did not find a generalization of this for $n\geq 3$.

Is there a nice formula for the maximal order of an element in $\mathrm{SL}(n,\mathbb{F}_q)$? If so, what is the proof or reference?

Remark: I would also be interested in the answer to the same question for other finite groups of Lie type, as well as the even characteristic case.

Sean Lawton
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  • For completeness: the maximal order of an element of GL(n,Fq) is q^n−1 - the cycle generated by that element is called "Singer cycle" - discussed in many papers and at MO e.g. Geoff Robinson here: https://mathoverflow.net/a/254509/10446 – Alexander Chervov Jun 18 '17 at 18:02

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Yes. It is shown in the paper

Darafsheh, M.R., Order of elements in the groups related to the general linear group., Finite Fields Appl. 11, No. 4, 738-747 (2005). ZBL1147.20043.

(Theorem 1) that the maximal order is $$\frac{q^n -1}{q-1},$$ except in the case $SL(2, p),$ where the maximum is $2p.$

Igor Rivin
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    Thank you very much Igor. This question came up yesterday in a meeting with undergraduates in my geometry lab. After thinking about the SL(2,q) case I optimistically guessed the above formula was the answer and that it was a known fact, but I would never have guessed it was a 2005 Theorem. – Sean Lawton Jun 17 '17 at 20:16
  • Of course now I wonder, given the relationship between maximal orders in GL, SL, PSL, and PGL, if 1\to H\to G\to Z(G)\to 1 or 1\to Z(G)\to G\to H\to 1 (for a finite group of Lie type G) does the maximal order of an element in H equal the maximal order of an element in G divided by |Z(G)|? It is probably too good to be true. – Sean Lawton Jun 17 '17 at 20:16
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    @SeanLawton Just because it was published in 2005 does not mean that it was not known to John Thompson is 1962 :) – Igor Rivin Jun 17 '17 at 20:18
  • Of course you are right :) I suppose I was expecting a book on finite groups from that time period as the answer, which is what surprises me. Thanks again! – Sean Lawton Jun 17 '17 at 20:25
  • It was probably known to Frobenius or anyone who knew how to study characteristic polynomials (and asked the question). – Andreas Thom Jun 18 '17 at 19:59
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    @AndreasThom Maybe, but that's no reason to take away credit from the guy who published the 2005 paper. Who knows how much of what we all published was known before... – Igor Rivin Jun 19 '17 at 00:28
  • @IgorRivin; that it true. – Andreas Thom Jun 19 '17 at 05:39
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    @IgorRivin: It is good to have a reference, but I was just thinking that Landau looked at the maximal order of elements in Sym(n) in 1903 because this was a difficult and interesting problem (and the solution relies on the prime number theorem). Those people must have thought about the same problem for linear groups, which is much easier. – Andreas Thom Jun 19 '17 at 07:25
  • @AndreasThom I completely agree - this is a very natural question. – Igor Rivin Jun 19 '17 at 13:23
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    Do you folks know of any reference for the maximal order of an element in $SL_2(\mathbb{Z}/p^k\mathbb{Z})$, where $p$ is a prime? – Alain Valette May 21 '18 at 17:09
  • Do you know any reference for the list of the order of elements in $GL(2,q)$? There are pieces of literature for the same in the case of symplectic and orthogonal groups though. Please let me know if there are any. – user300 Jul 08 '22 at 11:11