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I have some basic abstract algebra knowledge (the usual groups/rings/fields).

Now I would like to study, in depth, presentations of groups and group actions. (either of which I have no knowledge)

Could someone please recommend to me books for this purpose?

(I am aware that I will most likely need two different sources (or more))

I tried to search Amazon for books on group theory but I couldn't really find a good match. There is Joseph Rotman's book on group theory and while it seems to have a bit of both, according to the reviews it is full of typos and it also contains a load of other topics.

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  • $\begingroup$ I don't know remember how much it has about presentations, but Dummit and Foote is good for groups and group actions. $\endgroup$ Commented Jan 14, 2015 at 8:16
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    $\begingroup$ These two topics are not very strongly related, so you should not expect to find a single source that covers both and nothing else! I completely disagree with your disparaging remarks about Rotman's book, which is one of the books on group theory that I recommend most strongly. For example, I know of no other book at that level that includes an accessible proof of the unsolvability of the word problem in finitely presented groups. $\endgroup$ Commented Jan 14, 2015 at 9:09
  • $\begingroup$ What books you followed in your first course of algebra? $\endgroup$ Commented Jan 14, 2015 at 9:18
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    $\begingroup$ The book "Groups, Graphs and Trees" by John Meier (£20.79, second hand, amazon.co.uk) is a lovely introduction to the theory of group actions. However, flicking through it, it perhaps assumes a basic knowledge of presentations (they are only briefly introduced on page 68). On the other hand, I am a big fan of Chapter 1 of the book "Combinatorial group theory" by Magnus, Karrass and Solitar (£9.99, second hand, amazon.co.uk). It is, perhaps, the ultimate in introductions to presentations. It is also one of the classic texts in geometric and combinatorial group theory. $\endgroup$ Commented Jan 14, 2015 at 9:54
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    $\begingroup$ There is also the book "Presentations of Groups" by D.L.Johnson, but I cannot recall if it talks about actions. It does talk about (co?)homology though, and it was the only place I could find anything about cyclically presented groups. (That is, about presentations of the form $\langle x_0, x_1, x_2, x_3, x_4; x_{i}x_{i+1\pmod5}=x_{i+2\pmod5}\rangle$. This one is cyclic of order $11$, but would be infinite for large values of $5$.) $\endgroup$ Commented Jan 14, 2015 at 9:58

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It seems to me that you would like to have a book which combines aspects of geometric group theory and algebraic group theory. Here I would recommend the book Introduction to Group Theory by Oleg Bogopolski.

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  • $\begingroup$ Thank you. I looked at the TOC and it looks advanced. Is it really an introduction like the title suggests? $\endgroup$ Commented Jan 14, 2015 at 10:20
  • $\begingroup$ Well, some parts are advanced, but you still have the basic things you need on an elementary level. $\endgroup$ Commented Jan 14, 2015 at 10:32

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