Groups and symmetry.

*(English)*Zbl 0663.20001
Undergraduate Texts in Mathematics. New York etc.: Springer-Verlag. xi, 186 p. DM 64.00 (1988).

As its title suggests, the book is an introduction to group theory through examples which are mainly of geometric nature. From the purely group-theoretical point of view, the content goes until the Sylow and Nielsen-Schreier theorems.

The author follows throughout the book the principle (suitable in case of a book aimed to beginners): “at each point do everything you can with the methods at hand”. As an illustration, an ad-hoc argument due to McKay is used to prove Cauchy’s theorem in a separate chapter, though later on a new proof is produced by using permutation groups.

As a bit of criticism: the first part of Sylow’s theorem is not stated in its general version which asserts that a finite group has subgroups of any possible p-order, though the machinery developed and a slight modification of the argument would produce this assertion. What is asserted is only the existence of the Sylow subgroups; the result of this treatment is that Sylow’s theorem does not appear explicitly as an extension of Cauchy’s one.

The second half of the title is illustrated generously; symmetry groups of various solids and patterns are presented not only to justify the introduction of new algebraic concepts, but also for their own sake. The following titles of chapters are significant in this respect: Plato’s solids and Cayley’s theorem, Finite rotation groups, The Euclidean group, Lattices and point groups, Wallpaper patterns, Trees and the Nielsen- Schreier theorem.

The main idea behind the writing of this book is to present groups “at work”, by acting on various systems of objects. A wise attitude on behalf of the author. Unfortunately, because of lack of time, this cannot be realized to the same extent in a spoken course. This is the reason for which I think that this book cannot be used as such for teaching a first course in group theory. Nevertheless, it is a valuable compendium of nice examples, aimed to deepen the understanding of both groups and symmetry. And - this for sure - the beginner will enjoy working independently through it!

The author follows throughout the book the principle (suitable in case of a book aimed to beginners): “at each point do everything you can with the methods at hand”. As an illustration, an ad-hoc argument due to McKay is used to prove Cauchy’s theorem in a separate chapter, though later on a new proof is produced by using permutation groups.

As a bit of criticism: the first part of Sylow’s theorem is not stated in its general version which asserts that a finite group has subgroups of any possible p-order, though the machinery developed and a slight modification of the argument would produce this assertion. What is asserted is only the existence of the Sylow subgroups; the result of this treatment is that Sylow’s theorem does not appear explicitly as an extension of Cauchy’s one.

The second half of the title is illustrated generously; symmetry groups of various solids and patterns are presented not only to justify the introduction of new algebraic concepts, but also for their own sake. The following titles of chapters are significant in this respect: Plato’s solids and Cayley’s theorem, Finite rotation groups, The Euclidean group, Lattices and point groups, Wallpaper patterns, Trees and the Nielsen- Schreier theorem.

The main idea behind the writing of this book is to present groups “at work”, by acting on various systems of objects. A wise attitude on behalf of the author. Unfortunately, because of lack of time, this cannot be realized to the same extent in a spoken course. This is the reason for which I think that this book cannot be used as such for teaching a first course in group theory. Nevertheless, it is a valuable compendium of nice examples, aimed to deepen the understanding of both groups and symmetry. And - this for sure - the beginner will enjoy working independently through it!

Reviewer: M.Deaconescu

##### MSC:

20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |

20H15 | Other geometric groups, including crystallographic groups |

20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |