Groups and Symmetry

by M.A. Armstrong

Groups are important because they measure symmetry. This text, designed for undergraduate mathematics students, provides a gentle introduction to the vocabulary and many of the highlights of elementary group theory. Written in an informal style, the material is divided into short sections, each of which deals with an important result or a new idea. Throughout the book, emphasis is placed on concrete examples, often geometrical in nature, so that finite rotation groups and the 17 wallpaper groups are treated in detail alongside theoretical results such as Lagrange's theorem, the Sylow theorems, and the classification theorem for finitely generated abelian groups. A novel feature at this level is a proof of the Nielsen-Schreier theorem, using groups actions on trees. There are more than 300 exercises and approximately 60 illustrations to help develop the student's intuition.

Genres: Mathematics, Physics

198 pages, Hardcover

First published February 27, 1988

Reviews

[Pratik Aghor · Rating 5 out of 5]

A good starting point for anyone wanting to learn group theory. Chapters are super-small, making the material quite digestible. Also, good geometric intuition, my favorite was the orbits of a group action. Highly recommended.

[Asalmon14 · Rating 2 out of 5]

Although I do like the pedagogical style of the book, the proof of the Nielson-Schreier theorem is incorrect in two places. First, while the statement of 28.2 is correct, the proof given by Armstrong is incorrect, since (looking at the last sentence) we might not be able to replace z by y without disconnecting the tree. Second, in the proof of 28.3, Armstrong never proves that every reduced word can be created by the process he describes in the second paragraph (a fact which is implicitly assumed in the last paragraph of the proof). Theorem 28.3 doesn't seem to be true at all--the cyclic group of order 2 acts freely on the path of length 2.