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In my undergraduate research project, I am going to study a paper on free products in division rings. To do this, however, I, of course, need to learn about free groups and free products.

Right now, the only reference I have is Rotman's "An Introduction to the Theory of Groups". Is this a good reference? Or is there a better book to get the intuition and the main theorems behind free groups?

Please have in mind that I am self-studying and that, being an undergrad, if it is possible to avoid too much Category Theory, it would be best.

Thanks in advance!

Shaun
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Gauss
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    At the risk of evangelizing: I understand wanting to avoid "too much" category theory, but a little bit of category theory (namely universal properties and adjoint functors) can really grease the wheels when it comes to understanding free groups and free products. – HallaSurvivor May 07 '21 at 21:50
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    If I may add a suggestion to Shaun's excellent one: Stillwell's "Classical Topology and Combinatorial Group Theory" is characteristically well written. The group theory in that book definitely exists in service to the topology, but it does a great job presenting the basics and providing some context for a lot of these constructions (which are, in my mind at least, inherently geometric). – HallaSurvivor May 07 '21 at 21:54
  • @HallaSurvivor Do you have any book recommendations for an introduction to Category Theory? – Gauss May 08 '21 at 13:15
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    There's a few good ones, and everyone does things in a slightly different order. I recommend skimming the first chapter of Awodey's book and Leinster's book (which has the benefit of being free) and seeing whose style you like more. Also, the first chapter of Aluffi's Algebra book (and, while you're there, Chapter II.5 on free groups) is a great practical introduction. – HallaSurvivor May 08 '21 at 13:41

1 Answers1

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Try Magnus et al.'s, "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations". It's quite a comprehensive treatment of free groups and free products, alongside other concepts.

The following question of mine might be of interest too.

Different ways of constructing the free group over a set.

Shaun
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    +1 This is an excellent, classical text which has stood the test of time. The exercises are, to quote my supervisor, "a work of art". – user1729 May 07 '21 at 20:48