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An Introduction to the Theory of Groups

Joseph J. Rotman
Springer Verlag
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics 148
BLL Rating: 

The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by
Michael Berg
, on
In the dark ages, i.e. the late 1970’s, when I was a student at UCLA, I had the good fortune of taking the graduate algebra course with the late Ernst Straus.  He was one of my favorite and most inspirational teachers: a brilliant mathematician who generally lectured without notes, exposing his students to living mathematics, as it were: his textbooks were secondary to his lectures.  But these textbooks were invariably superb: they always supplemented Straus’ lectures perfectly and given the level at which he pitched his courses, they needed to be read and studied.  So it was, for example, that his introductory course in number theory was accompanied by Hardy and Wright, Introduction to the Theory of Numbers, and when I attended his seminar on transcendental number theory, we used Siegel’s Transcendental Numbers, as well as Alan Baker’s gorgeous but dense Transcendental Number Theory.  Straus’ taste was classical and the textbooks he used in his courses were classics.
Thus, in the aforementioned graduate algebra course, when we got to group theory, Straus had us study the book under review, J. J. Rotman’s An Introduction to the Theory of Groups, and for me. it was, as the song says, the start of something big.  I was still quite a rookie at that point, but my previous (undergraduate) exposure to group theory had come via another, but very differently flavored, classic, Herstein’s Topics in Algebra.  I was struck by a very different approach to the subject on Rotman’s part, and, granting the justifiably famous pedagogical impart of Herstein’s book, I much preferred Rotman’s approach.  How to describe the difference?  Perhaps it is proper to say that Herstein’s focus is for the (very) serious beginner to get covered in the mathematical dirt that accompanies digging into the mathematical soil --- a sine qua non, of course --- but Rotman’s focus is structure, beautiful algebraic structure.  For me the sequence was perfect: Herstein first, then Rotman.  I felt like the material I had learned from Herstein’s Topics was all but refashioned, architecturally, and then built into a coherent edifice.  Perhaps it was also a matter of evolving mathematical maturity, but it certainly hit me at the right time and has stayed with me ever since.
Indeed, I became quite a fan of Rotman’s work: his (at nearly 1000 pages) Homeric work, Advanced Modern Algebra, is among my very favorite books, one I have used time and again in connection with my own research, and his Introduction to Homological Algebra is, among other things, a fabulous source for spectral sequences.  And it all started with my introduction to his works in the 1970's.  My claim is that even after almost a half-century (it first saw the light of day in 1965) the book under review is a fabulous source from which to learn group theory all the way up to the graduate school level.  And then I hope the student will develop the same taste for all things Rotman that I have.


Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.


Groups and Homomorphisms.- The Isomorphism Theorems.- Symmetric Groups and G-Sets.- The Sylow Theorems.- Normal Series.- Finite Direct Products.- Extensions and Cohomology.- Some Simple Linear Groups.- Permutations and Mathieu Groups.- Abelian Groups.- Free Groups and Free Products.- The Word Problem.- Appendices: Some Major Algebraic Systems.- Equivalence Relations and Equivalence Classes.- Functions.- Zorn's Lemma.- Countability.- Commutative Rings.- Bibliography.- Notation.- Index.