From a teaching perspective, one of the first choices an instructor intending to teach an undergraduate course in abstract algebra has to make (or, depending on the size of the department, has made for them) is whether to cover groups or rings first. After that, the textbooks supporting these choices have almost no overlap. Sibley’s Thinking Algebraically, according to its back cover, “succeeds in combining the advantages of rings-first and groups-first approaches while avoiding the disadvantages.” To put Sibley’s approach in the context of other texts, we might call it “morphisms almost immediately, then groups-first.”

The first two chapters are an absolute whirlwind of definitions. On pages 18-21, the reader is exposed to groups, closure, cancellation, equation solving, exponents, commutativity, “a negative times a positive is a negative” (proof left as an exercise for which there is a solution in the back of the book), “a negative times a negative is a positive” (proof left as an exercise), ring, unity, unit, field, and a brief proof that Q(sqrt(3)) is a field (the first extended example). On pages 30-35, the reader finds: permutation groups, cyclic and dihedral groups, isometries, symmetry groups, the division algorithm, congruence mod n, equivalence relation, the integers mod n form a commutative ring. Soon after (page 52-53), the reader meets isomorphisms and finds that the nth roots of unity are isomorphic to the integers mod n. The exercises that support these early sections have a similarly wide scope and ambition: among other things, the reader finds Egyptian fractions and the Babylonian method for completing the square in 1.1, matrix rings and polynomial rings and function spaces in 1.2, Cayley tables and UPC codes in 1.3… The general early feeling of zealous and breathless enthusiasm sometimes hampers the presentation of critical details. In Section 2.4, titled “Homomorphisms” but also covering kernels, cosets, and Lagrange’s theorem, here are the sentences immediately following the definition of kernel: “By part (ix) of Theorem 2.4.1, the kernel of a group homomorphism is a subgroup since the identity of the image forms a subgroup. We use matrices to represent linear transformations and also systems of equations to solve. The solutions of the homogeneous system Mx = 0 form the kernel ker(M) when we think of M as a linear transformation.” For a reader who’s never previously been exposed to the definition of a kernel, this is quite a heavy lift.

The style settles down significantly in Chapter 3, which discusses cyclic groups in careful detail, shows how the fundamental theorem of finite abelian groups is used while pushing the proof to an appendix (a good idea!), gives multiple pictorial examples of Cayley digraphs, uses graphical representations of molecules to illustrate group actions, and has one of the clearest illustrations of a quotient group (Figure 3.23, page 153) that I’ve ever seen. The exercises associated with this chapter are significantly more grounded and computational, often reflecting and deepening the examples worked in the sections. Chapter 4 moves quickly and coherently through the big ideas of ring theory, ending with well-illustrated asides on Grobner bases and polynomial dynamical systems. In Chapter 5, brief sections on vector spaces (with computational exercises often done over finite fields to illustrate the benefit of abstraction) and linear codes are an excellent way to gently reintroduce computational linear algebra before returning to “the main plot” with algebraic extensions, geometric constructions and their impossibility, and the fundamental theorem of Galois theory, culminating with a detailed example of a quintic whose roots cannot be solved by radicals.

Chapters 6 and 7, nearly a quarter of the length of the book, are devoted to individual self-contained sections on topics in group and ring theory. These sections can be read in any order once the content of Chapters 1-5 is completed, and each comes with a significant number of computational exercises (something often missing in “special topics” sections!). These sections include fairly gentle introductions to symmetry groups, matrix groups, Sylow theorems, lattices, boolean algebras, and more. A group of students in their first week of an REU on one of these topics could easily use one of these sections as a jumping off point before diving into primary sources. Further, throughout the book are surprisingly detailed (easily half a page each) biographies of algebra’s major historical figures. There are so many of these that the book ends with a two-page index of names: this is a thoughtful inclusion, and mathematical history does not feel like an afterthought in this book. Also, it’s worth noting that the cover – a photograph of a glass octahedron embedded inside an icosahedron, made as part of a student project by the author – is stunningly beautiful.

Overall, the title

*Thinking Algebraically* might suggest that this book would be a careful introduction to the subject with a focus on detailed and gentle proofs, something in the spirit of an updated Fraleigh’s

*First Course in Abstract Algebra*. This is not that book, especially not in the first two chapters. However, Thinking Algebraically does have a lot to offer for students who can skim the first two chapters and are ready to dive into senior-level projects which often require a significant amount of individual attention and curation by motivated instructors. The second step (something beyond a gentle introduction, but not quite graduate-level in tone) is often a neglected one, and

*Thinking Algebraically* is full of potential second steps.

Steve Balady is Assistant Professor of Mathematics at California State University San Bernardino. Their dissertation is in algebraic number theory, and their current research work focuses on student identity and belonging in the undergraduate math classroom.