I would please like some recommendations for an introductory level book on combinatorial group theory, by which I mean a group theory book which places emphasis on generators and relations and free groups, and then discusses common concepts such as quotient groups in terms of these. Thank you.
I would suggest the book
- Groups, Graphs and Trees an introduction to the geometry of infinite groups by John Meier.
This is an excellent introductory text. It is well written, covers a broad range of topics in geometric and combinatorial group theory, and contains lots of examples (every second chapter is a study of an example). Also, it is modern (2008) - the other suggested texts are all 80s and earlier! (The other books are, of course, still very relevant, but Meier's book allows you to see where the subject is today, as opposed to in the pre-Gromov days.)
That said, I do not believe that you can survive in the world of combinatorial group theory without reading Magnus, Karrass and Solitar. So Meier's book is actually my secondary recommendation. Read Meier so that when you read Magnus, Karrass and Solitar you will understand it better.
As mentioned, Presentations of Groups by D.L. Johnson is nice.
Try also these books:
Topics in the Theory of Group Presentations by D.L. Johnson
Combinatorial Group Theory by Roger C. Lyndon and Paul E. Schupp
Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, by Wilhelm Magnus, Abraham Karrass, Donald Solitar
I would recommend "Presentations of Groups" by D.L. Johnson. It is good as an introduction to free groups and group presentations, but does assume a basic knowledge of group theory. Since a group defined by a presentation is defined as a quotient group of a free group, you need to have some basic familiarity with quotient groups and the isomorphism theorems before starting to learn about group presentations.
I would also recommend the book Trees of Jean-Pierre Serre, which has a highly original and elegant approach to combinatorial and geometrical group theory.