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What is a good book to start a journey in the field of infinite group theory ? I have already taken a first course in algebra where we studied the most important (finite) algebraic structures and I'm taking the second course so I'm used to the basic tools of abstract algebra, however Infinite groups (except for $\mathbb{Z}$ of course) has only been cited as examples and never studied in details so I'd like a text to start the topic but that also focuses on it (without the "finite group theory" part). I'd like a book as general as possible but if I have to choose a particular kind of groups I guess Linear Groups are a good point to start (as I already encountered them in other courses).

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    $\begingroup$ There isn't really such a thing as general group theory/infinite group theory in a meaningful way (there are not really interesting theorems at that generality). $\endgroup$ – Paul Plummer Apr 3 '17 at 21:35
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    $\begingroup$ Although currently there does not seem to be a field of study named what you ask about, I do seem to recall a small monograph by a group theorist (ack, whose name eludes me at this moment) from the 1960s or so, called "infinite abelian groups". I once did own a copy, but now cannot find it. I confess I did not find much interest in it, and only looked at it so long ago that I would have been unable to "rewrite it" to make it "more conceptual" (to my own way of thinking, of course). But google "inf. ab. gps" on Amazon and see what comes up... $\endgroup$ – paul garrett Apr 4 '17 at 0:57
  • $\begingroup$ Start by studying groups like ${\Bbb Z}_2*{\Bbb Z}_2$ or ${\Bbb Z}_2*{\Bbb Z}_3=PSL_2({\Bbb Z})$, then you will be pushed to study Free Groups, Free Products, Amalgamated Products, Bass Serre theory, etc. $\endgroup$ – janmarqz Apr 4 '17 at 22:01
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    $\begingroup$ @paulgarrett, Kaplansky: Infinite abelian groups. $\endgroup$ – lhf Apr 4 '17 at 23:57
  • $\begingroup$ Thx, @lhf! :) ... $\endgroup$ – paul garrett Apr 5 '17 at 0:11
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As said by @PaulPlummer, there is no general theory of infinite groups.

There are nonetheless some special theories that are still quite broad and are very important. They do tend to tie in with other branches of mathematics, though.

Beyond your idea of linear groups I'll make just a couple of suggestions:

  • Lie groups (with close ties to differential geometry).
  • Coxeter groups (these are rather special and yet are a pre-requisite to a deeper understanding of both linear groups and Lie groups).
  • Combinatorial group theory (which is closely related to topology via the fundamental group of a topological space).
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  • $\begingroup$ Do you have any book recommendations on this topic? $\endgroup$ – Renato Faraone Apr 4 '17 at 5:50
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    $\begingroup$ For Coxeter Groups, Humphrey's book "Reflection groups and coxeter groups" is a good intro. I also recommend the first chapters of Mike Davis' book "The geometry and topology of Coxeter groups"; in fact I recommend the whole book but it's quite a lot to chew at first. $\endgroup$ – Lee Mosher Apr 4 '17 at 12:16
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    $\begingroup$ The classic books on Combinatorial Group Theory are one by Lyndon and Schupp, and one by Magnus, Karass, and Solitar. But this field has undergone a somewhat revolutionary change over the last few decades, and has morphed into the field of Geometric Group Theory; for some books see the answer of @AndyPutman in this question mathoverflow.net/questions/3858/… $\endgroup$ – Lee Mosher Apr 4 '17 at 12:20
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    $\begingroup$ For Lie Groups there's a zillion choices and I do not have a favorite, to be honest. My own education in Lie Groups was somewhat shallow, with little bits from here and from there. Take a look at this question for some information: math.stackexchange.com/questions/194419/… $\endgroup$ – Lee Mosher Apr 4 '17 at 12:22
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I can recommend the two volumes of Derek Robinson Finiteness Conditions and Generalized Soluble Groups (Part 1 and Part 2) which are probably discontinued, but possibly available in a math library near you. They are an excellent source to start with. Part 2/Contents I found on-line in .pdf format. Note that a lot of research on finite groups has been inspiring the research on infinite groups. That is why certain classes or even varieties of groups are studied. To this end also the book of Hanna Neumann Varieties of Groups makes an interesting read. Finally, you should certainly have a look into Jean-Pierre Serre's book Trees. Serre is one of the greatest mathematicians of our time and has written here a very original approach to a lot of infinite group theory. Enjoy!

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Kaplansky's Infinite abelian groups is a classic. Read a review.

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I would like to stress my commentary in the sense of the kind of starting things that provoke those concepts that pervade the branch of infinite groups. For example we have:

If $G$ is a countable group then it can be embedded in a group with two generators.

This theorem is from the 1970s and signals about the borning-techniques still in evolution on this huge branch.

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