Textbook recommendation on group theory for graduate. During studies I took a course on abstract algebra that involved elements of group, ring and field theories. Unfortunately the course was really basic and currently I'm searching for a book that maybe expand it a little bit more. Are there any good books that talk about these subjects focusing on more "advanced" topics? And by "advanced" I mean book for someone that already knows the basic definitions and theorems but want to explore that field of math a little bit more.
I'm primary interested in group theory - in particular group actions on a sets.
The basic idea is to learn something that would be useful in other parts on mathematics - for example in topology or geometry.
 A: It really depends on whether you want something a little more encyclopedic, or something with a different goal.
Here are some suggestions:
Encyclopedic. These are books that have a lot of stuff on a lot of topics, but generally do not delve very deep on any particular subject. They will have pointers to further work, though:

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*Joseph Rotman's An Introduction to the Theory of Groups; 4th Edition is from Springer. It starts pretty much from zero, but you should learn some new stuff from it about $G$-sets, and maybe even the Sylow Theorems. Once you hit Chapter 5 (of 12), it will probably be mostly new to you.


*Lang's Algebra, revised third edition from Springer. The encyclopedia of Algebra. In my opinion, not a great book to learn from, but a great reference. It is not exclusively Groups, though, and quickly veers into Fields and other stuff.


*Hungerford's Algebra is similar to Lang's.


*Derek Robinson's A Course in the Theory of Groups, second edition, from Springer. Starts with the basics, but it hits Free Groups and presentations in Chapter 2 (of 15) and continues from there. It is a bit more focused on infinite groups rather than finite groups.


*Kurosh's The Theory of Groups. There are two volumes, link is to volume 2, and I spotted a "free PDF" on a google search.
Somewhat more advanced, with a particular point of view


*Peter Neumann, Gabrielle Stoy, and Edward Thompson's Groups and Geometry from Oxford Science Publication, is more focused on groups as measurements of symmetry, with a good chunk on actions and the connection between groups and geometry, as the title might imply.


*I. Martin Isaacs, Character Theory for finite groups is the book to learn group characters from (it is related to representation theory). From Dover, it is fairly inexpensive.


*I. Martin Isaacs, Finite Group Theory, is a kind of "A second course in group theory". It starts assuming you know basic group theory and goes into the Sylow Theorems (with a couple of interesting extensions you don't usually see). The goal of the book is to give a representation-free proof of Burnside's $p^{\alpha}q^{\beta}$ theorem. It is published by the AMS in their Graduate Studies in Mathematics series.


*Magnus, Karrass, and Solitar's Combinatorial Group Theory is the classic book on combinatorial group theory.
Specific to group theory in topology
You mention applications of group theory to topology. There are a couple of books that deal with that directly.

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*From a more topological point of view, we have John Stillwell's Classical topology and combinatorial group theory.


*For one of the classics, Max Dehn's Papers on Group Theory and Topology. Keep in mind that these are the papers he published. Influential, but possibly hard going to begin with.
There are a few other books that might contain interesting stuff depending on your interest. I'm afraid I don't have one that deals specifically with group actions. Some of the older books might contain interesting stuff that is not done much any more (Marshal Hall's classic book "The Theory of Groups" has a bunch of stuff on the Burnside problem and Jordan's theorem on multiply transitive groups). Some have specific topics in mind, such as Susan McKay's book on $p$-groups, or Hanna Neumann's classic book Varieties of groups, J-P Serre's Trees for groups acting on trees, etc., but those are the kind of books you should search for later, after you've decided you want to pursue this. If you want to work in abstract algebra, you'll probably want to get Lang or Hungerford at some point.

My recommendation if you are not wedded to looking at group actions is Isaacs' Finite Group Theory. If you definitely want to look at group actions, look at either Neumann, Stoy, and Thompson's book, or the relevant chapters in Rotman. But that's just my opinion, and I'm not a very big "geometry" guy, so someone else might be able to better advice you there.
A: If you are looking for a book to read as a supplement, not as a textbook for preparing for graduate study, then I recommend Groups, Graphs, and Trees by John Meier.  It's lovely book with many interesting examples.  It does assume a first course in group theory, but it's not important that you remember any of the theorems---just that they are not ideas you are seeing for the first time.  I've used this book for reading courses and to as a source of lectures on specific topics, e.g. Thompson's group F, the lamplighter groups.  The emphasis is on group actions.  Professor Meier's so-called "Cayley's Better Theorem" ought to be part of any course on group theory with an emphasis on group actions.  This is the kind of book you read to learn more examples and because you are curious & hungry for beautiful ideas, not because you are trying to learn the formalities of a graduate level approach to algebra.
If you are looking more for a preparation for graduate studies, then the textbooks recommended in the comments are more suitable.
Rotman's book is very good for learning more group theory, but you have to read quite a few chapters before you see the connections to geometry and topology.  I learned a lot from this book, and I still use it as a reference from time to time.
A recent book that I like, but which may be a little terse, depending on your taste, is Introduction to Group Theory by Bogopolski.
