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Problems in Group Theory

J.D. Dixon
Dover Publications
Publication Date: 
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This is a useful collection of problems (with complete solutions) in group theory. It is intended as a supplement to a group theory text; it is not a standalone problem course such as Pólya & Szegő’s Problems and Theorems in Analysis. The book covers a wide range of topics and would be suitable for supplementing a first or second course in group theory. It is a Dover 1973 corrected reprint of a work originally published in 1967 by Blaisdell, and was reprinted again in 2007.

For the most part the problems are sidelines or examples, and are not the main results of group theory (the relevant main results are summarized with references at the beginning of each chapter). The author (p. vi) characterizes the problems as “challenging and accessible”, and I agree. They are especially good at requiring you to pull together facts from two or three areas of group theory to reach the conclusion, helping you integrate your knowledge. Working through the problems will teach you a lot about group theory. The solutions run about a quarter of a page each (with a lot of variation), so they’re not extremely difficult or especially easy, once you’ve hit upon the right facts to use. On the average they are probably a little harder than you would find on a Ph.D. qualifying exam.

The book provides a good mix of abstract and concrete problems. Some concrete examples are (problem 2.25) to show there is no simple group of order 56 and (problem 11.2) to work out the character table of the alternating group \(A_4\). The concrete problems are not simple drill and require some ingenuity and a good understanding of groups.

Despite its age, the book is still up to date. The only thing that might be missing by present-day standards is commutative diagrams with a category approach. One limitation of the book is that it presents pure group theory, in the sense that there are no applications to other parts of mathematics or other subjects. For example, there are chapters on normal series and on solvable groups, but no mention of Galois theory. There is some discussion of actions and orbits, but no mention of using them for combinatorial problems. But within the boundaries of group theory it does very well.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

List of Symbols; Introduction
1. Subgroups. Problems. Solutions
2. Permutation Groups. Problems. Solutions
3. Automorphisms and Finitely Generated Abelian Groups. Problems. Solutions
4. Normal Series. Problems. Solutions
5. Commutators and Derived Series. Problems. Solutions
6. Solvable and Nilpotent Groups. Problems. Solutions
7. The Group Ring and Monomial Representations.. Problems. Solutions
8. Frattini Subgroup. Problems. Solutions
9. Factorization. Problems. Solutions
10. Linear Groups. Problems. Solutions
11. Representations and Characters. Problems. Solutions
References; Index of Terms