Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $G$ be a finite group of Lie type in characteristic $p$. When is the Sylow $p$-subgroup of $G$ cyclic?

share|improve this question
7  
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
3  
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
2  
I don't know how I use this. But I will try to write better –  gauss Mar 5 '12 at 16:57
1  
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

1 Answer 1

up vote 12 down vote accepted

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?

share|improve this answer
    
@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 –  gauss Mar 5 '12 at 16:54
    
@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. –  gauss Mar 5 '12 at 17:04

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.