Let $G$ be a finite group of Lie type in characteristic $p$. When is the Sylow $p$-subgroup of $G$ cyclic?
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7$\begingroup$ To answer the question in the title: $|GL(2,p)|=p(p-1)(p^2-1)|$. Hence the Sylow p-subgroup of $GL(2,p)$ is cyclic. $\endgroup$– RalphCommented Mar 5, 2012 at 13:08
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What is meant by "finite group of Lie type" needs to be made precise. But at least the simple groups of Lie type in characteristic $p$ with a cyclic Sylow $p$-subgroup are easy to specify: these are the groups $\text{PSL}(2,p)$ with $p>3$ along with one twisted group usually denoted $^2 \text{G}_2(3)'$ with $p=3$ (which is isomorphic to $\text{SL}(2,8)$). Of course there are also some closely related non-simple groups of Lie type including a few very small groups with $p=2$
This is summarized on page 74 of my 2005 Cambridge Univ. Press book Modular Representations of Finite Groups of Lie Type along with what I hope are sufficient references to the scattered literature.
P.S. Whether or not a finite group has a cyclic Sylow subgroup (for some prime) usually comes up in two contexts: blocks with a cyclic defect group (Brauer, Dade) and finite representation type for finite dimensional algebras including group algebras. Are there other motivations?
answered Mar 5, 2012 at 13:38
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$\begingroup$ @jim thanks for your answer. My question is related to trivial intersection property. And a maximal curve (curves that attains hasse- weil bound) defined over a finite field F(q), has automorphism group G whose sylow p-subgroup has this property $\endgroup$– gaussCommented Mar 5, 2012 at 16:54
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$\begingroup$ @jim I have your book ''modular representations of finite group of lie type'' and I will look at this page.I did not say in my question but I know of course the group PSL(2,p) is an example of the property mentioned in my question thanks again. $\endgroup$– gaussCommented Mar 5, 2012 at 17:04