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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?

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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.