# Can the Sylow p-subgroup of a finite group of Lie type be cyclic?

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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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. – Ralph Mar 5 '12 at 13:08
Next time please use proper TeX, English, spelling, and punctuation. There were more than 10 mistakes in your 2 lines of text. – GH from MO Mar 5 '12 at 15:31
I don't know how I use this. But I will try to write better – gauss Mar 5 '12 at 16:57
1. For using this site, I recommend to read mathoverflow.net/faq, in particular mathoverflow.net/faq#latex 2. For learning TeX there are many good books, check out en.wikipedia.org/wiki/TeX 3. For English (in particular, for spelling and punctuation) you can take a language course, hire a private teacher, or just use the resources from the web. 4. Many mistakes can be avoided simply by paying attention to what you write. For example, the sentence "But I will try to write better" should end with a dot ("."). – GH from MO Mar 5 '12 at 18:38

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 $p=2$