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in Dummit and Foote , the notion of presentation is introduced in section 1.2 which talks about dihedrial group of order $2n$.

and after this , it was rare to talks about presentation throw the exercises or the material of the sections . but suddenly in chapter 4 , it asks to create or make a presentation of particular groups , and i don't know the right methods and strategies - if any exists ! - which i can follow

so , any help with that ? i think that presentations and relations play fundamental role in the advanced levels of the study of group theory , so it's good to develop my skills with it from now and on " just a feeling ! am i wrong ? "

FNH
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    I think you'd better focus on one of the exercises the book wanted you. For example, does the book want you to create a presentation about $D_6$, $Q_8$ or $D_6\times Q_{12}$? Bring the target group up here and wait to get the hints or answers. :-) – Mikasa Jun 05 '13 at 07:48
  • The easier examples encountered in exercises are often finite solvable groups, and there are systematic ways of writing down presentations of them. As Babak said in his comment, it would be easier to answer if you suggested some examples. – Derek Holt Jun 05 '13 at 10:30

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There is no general strategy that you can follow to obtain a short presentation for a given group. Some things to look for are elements of particular orders that you can easily identify in the group. If you have an element of order $k$, then add a generator $g$ and a the relation $g^k=e$. Next, among such elements, or powers of such elements, look for conjugation relations and similar things. That is, suppose you have two generators $g,h$ in your presentation and you want these to correspond to two elements in the group that you have. Then check for their orders and the way the multiply with each other and with powers of each others. Among these try to find those that look like the most important ones (those from which the others follow). For small groups this will soon enough lead to a presentation after enough trial and error and your intuition will improve. However, this presentation may be shortened by noticing that some relations are redundant as well as by choosing different generators.

Ittay Weiss
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  • great , but how can i know that the relations which i get are enough to present all the information which the group has ? – FNH Jun 05 '13 at 08:33
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    you look at the presentation you get and you try to derive all the relations in the group. You also need to show you did not include too many generators (i.e., that you don't end up with too many elements). This isn't a trivial matter and for each group will require careful attention and perhaps some tricks. Try, keep trying, try harder. – Ittay Weiss Jun 05 '13 at 08:36