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Conider the following abstract description for the Quaternion group:

$$\langle x,y\mid x^{4}=1,x^{2}=y^{2},y^{-1}xy=x^{-1}\rangle$$

This description is called a presentation of the Quaternion group via generators and relations.

I am looking for a modern and thorough exposition for presentations of groups because I think it is a fascinating area about which I would like to know more. Are there any books or comprehensive online sources that you could recommend?

Thank you for your ideas!

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    $\begingroup$ Dear Moritz, almost every book on group theory has a chapter on presentation of groups, free groups, free products of groups and so on :) Perhaps you like my lecture notes, in particular chapter $4$ and its references. $\endgroup$ – Dietrich Burde Dec 16 '18 at 16:38
  • $\begingroup$ @Dietrich Burde: Thank you for your kind response. I know that books about group theory have often small chapters (in general about 10 pages long) about this topic or treat it as a secondary subject. I was looking more for a "monograph" about presentations - if such a thing exists. I also found lecture notes from a course by Derek Holt about the theme but they were taken by a student and were full of errors. So, my question is about a good personal recommendation that someone could give me: comprehensive, modern and - if possible - error free. A book is good, an online source even better. $\endgroup$ – Moritz Dec 17 '18 at 15:39
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    $\begingroup$ Have you looked at Presentation of groups by Johnson, or any book on/titled Combinatorial group theory? $\endgroup$ – Paul Plummer Dec 17 '18 at 17:48
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    $\begingroup$ Possible duplicate of Combinatorial group theory books $\endgroup$ – Paul Plummer Dec 17 '18 at 20:20
  • $\begingroup$ While I voted to close as duplicate it is worth pointing out that combinatorial group theory is very influenced nowadays by geometric group theory, so it is also worth looking into that. Check out this question for that $\endgroup$ – Paul Plummer Dec 17 '18 at 20:23
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The answer occurs in the comments above. I summarize

  • "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations" by W. Magnus and others, 444 pages, ISBN 0486438309, 2004-reprint of 1976-edition
  • "Presentations of Groups, 2ed" by D.L. Johnson, 232 pages, ISBN 0521585422, 2008-reprint of 1997-edition
  • "Topics in the theory of group presentations" by D. L. Johnson, 311 pages, ISBN 9780521231084, 2008-reprint of 1980-edition
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