A new book by Jean-Pierre Serre always deserves attention. This one is actually new and old. It is based on a course given by Serre in 1978–79. A typeset version of the notes was prepared in 2004 and eventually posted to the arXiv. Garving K. Luli and Pin Yu translated the original notes, and then Serre “revised and expanded them (by a factor of 2) for the present publication.” The expansion includes more references, exercises, and two new chapters, so even those who can read the original notes will want to take a look at this book.

While the title presents this as “an introduction,” the reader should understand it as an introduction to the theory of *finite* groups aimed at graduate students and mathematicians. Thus, Serre assumes something like the contents of a graduate course in algebra. He also points out many connections to other areas of mathematics, assuming at least some background knowledge in those areas.

When it comes to the theory of finite groups, there are really two quite distinct streams of research. On the one hand there is the theory of finite *simple* groups, culminating in the classification theorem, while on the other hand are such topics as solvable and nilpotent groups, the extension problem, etc. While the classification theorem is mentioned here (indeed, with some concern about its ultimate correctness), the main focus is on the other stream of ideas. So there are chapters on solvable and nilpotent groups, group extensions (and cohomology), Hall subgroups, Frobenius groups, the transfer homomorphism, and so on. Throughout Serre notes connections and applications to other areas of mathematics.

The two added chapters are particularly nice. The first is an account of two important theorems about finite matrix groups: Minkowski’s theorem about finite subgroups of \(\mathrm{GL}_n(\mathbb{Q})\) and Jordan’s theorem on finite subgroups of \(\mathrm{GL}_n(\mathbb{C})\). This is very nice stuff, and Serre’s account is (as usual) excellent. The final chapter on “small groups” is literally that: a collection of interesting facts about groups of small order. The first section gives many unexpected isomorphisms, from the well-known \(S_3 \cong D_3 \cong \mathrm{GL}_2(\mathbb{F}_2)\) to the more exotic \(A_8\cong \mathrm{SO}_6(\mathbb{F}_2)\cong \mathrm{SL}_4(\mathbb{F}_2)\). The second section discusses embeddings of \(A_4\), \(S_4\), and \(A_5\) into \(\mathrm{PGL}_2(\mathbb{F}_q)\).

And the exercises! Anyone who has attempted to solve the problems in Serre’s books knows that he is a master at finding exercises that are simultaneously difficult, interesting, and instructive. He has added about 150 of them to this book, which means that anyone who works through them will learn a lot.

Also not to be missed are the notes at the end of the index, on “ambiguous terminology.” Here Serre points out, for example, that a “set of generators of \(G\)” could well mean a set of elements, each of which generates \(G\). Luckily, no one uses it that way. On the other hand, as he also notes, the word “positive” is used in different senses: it can mean \(\geq 0\) or \(\gt 0\). “In this book, positive is only used in a statement if that statements is true with both interpretations,” says Serre, which is a particularly sly way of choosing one of the meanings.

Not all great mathematicians are also great writers. Jean-Pierre Serre excels at both things. *Finite Groups* is another wonderful book from one of the great expositors of our time.

Fernando Q. Gouvêa once attempted to solve all the exercises in Serre’s *Local Fields*. He survived, and is now Carter Professor of Mathematics at Colby College.