# Maximal torus and application in prime graph [closed]

I am studying papers about " Prime graph" , for example "Prime graph components of finite groups" [ williams], " Groups with complete prime graph connected components" [ Lucido and moghaddanfar]. But I do not understand some of scientific terms in papers to relate this matter. For example, concepts like torus, maximal tori. Can someone explain these terms? How these concepts are used in these articles?

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## closed as off-topic by Gjergji Zaimi, Andy Putman, Andrey Rekalo, Chris Godsil, Todd Trimble♦Jun 30 '13 at 1:56

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Gjergji Zaimi, Andy Putman, Andrey Rekalo, Chris Godsil, Todd Trimble
If this question can be reworded to fit the rules in the help center, please edit the question.

They are talking about maximal tori in finite groups of Lie type. See e.g. the books by Carter (ams.org/mathscinet-getitem?mr=794307) or Malle-Testerman (ams.org/mathscinet-getitem?mr=2850737). –  Francois Ziegler Jun 28 '13 at 17:58
Thank you for your response –  mousavi Jun 29 '13 at 6:28

## 1 Answer

We know that each torus is a maximal abelian subgroup and so the prime divisors of $|T|$ are mutually adjacent in the prime graph of $G$.

So knowing the order of maximal abelian subgroups of a simple group give many adjacency in the prime graph.

Only one need to determine the adjacency between the characteristic of the field related to the simple group of Lie type to determine the prime graph of a simple group.

I hope that it be helpful.

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Thank you, very good answer.How can I find the order of maximal abelian subgroups of a simple group? Where can I find information about simple groups that are easy to understand. –  mousavi Jun 29 '13 at 6:06
You can see the books where introduced by Francois Ziegler in the previous section. –  BHZ Jun 29 '13 at 6:14
Also you can use the Wilson's book about the simple group which is a very nice text. –  BHZ Jun 29 '13 at 6:14