What are some books that discuss transformation groups (or permutation groups) before abstract groups?

Some quotes to motivate the question:

from V. I. Arnold, 'On Teaching Mathematics':

What is a group? Algebraists teach that this is supposedly a set with two operations that satisfy a load of easily-forgettable axioms. This definition provokes a natural protest: why would any sensible person need such pairs of operations? "Oh, curse this maths" - concludes the student (who, possibly, becomes the Minister for Science in the future).

We get a totally different situation if we start off not with the group but with the concept of a transformation (a one-to-one mapping of a set onto itself) as it was historically. A collection of transformations of a set is called a group if along with any two transformations it contains the result of their consecutive application and an inverse transformation along with every transformation.

This is all the definition there is. The so-called "axioms" are in fact just (obvious) properties of groups of transformations. What axiomatisators call "abstract groups" are just groups of transformations of various sets considered up to isomorphisms (which are one-to-one mappings preserving the operations). As Cayley proved, there are no "more abstract" groups in the world. So why do the algebraists keep on tormenting students with the abstract definition?"

from Miles Reid, 'Undergraduate Commutative Algebra', p.145:

No subject has suffered as badly from the insistence on the abstract treatment as group theory. When I was a first year undergraduate in Cambridge in 1966, it had been more or less settled, presumably after some debate, that the Sylow theorems for finite groups were too hard for Algebra IA; since then, the notion of quotient group, and subsequently the definitions of conjugacy and normal subgroup have been squeezed out as too difficult for the first year. Thus our algebraists have cut out most of the course, but stick to the dogma that a group is a set with a binary operation satisfying various axioms. Groups can be taught as symmetry groups (geometric transformation groups), and the abstract definition of groups held back until the student knows enough examples and methods of calculation to motivate all the definitions, and to see the point of isomorphism of groups.

The schizophrenia between abstract groups and transformation groups comes to the surface in some amusing quirks – for example, the textbooks that define an "abstract group of operators", or the students (year after year) who insist that the binary operation $G \times G \to G$ on a group should satisfy closure under $(g_1, g_2)\mapsto g_1 g_2$ as one of the group axioms.

[Note that the Cambridge syllabus has changed since this: there's now a dedicated first-term 'Groups' course that covers many of the things Reid mentions.]

  • $\begingroup$ The subject of Geometric Group Theory has oodles of books, and they're all from the perspective of group actions / permutation / symmetry groups. Perhaps I'm biased from all my exposure to the 3-manifolds literature but my impression is that almost all modern group theory texts use this perspective as the primary perspective. $\endgroup$ – Ryan Budney Sep 7 '13 at 10:18
  • $\begingroup$ @RyanBudney: The question is about an introductory course. $\endgroup$ – Mark Sapir Sep 7 '13 at 11:09
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    $\begingroup$ @Marius: I do not know about Miles Reed, but most of Arnold's rants about education meant to be just provocative jokes. He also said, if I remember correctly, that most graduates from ENS in France cannot add fractions (because they are too busy studying schemes). $\endgroup$ – Mark Sapir Sep 7 '13 at 11:22
  • $\begingroup$ I seem to remember that "Groups and Geometry" by Peter Neumann, et al, places a lot of emphasis on groups as "acting groups" (that is, studying groups by what they do rather than what they are). But it's been a while since I looked through it. I'll try to take a look on Monday. $\endgroup$ – Arturo Magidin Sep 7 '13 at 21:15

An excellent book for the beginners is

MR0917939 Nikulin, V. V.; Shafarevich, I. R. Geometries and groups. Translated from the Russian by M. Reid. Universitext. Springer-Verlag, Berlin, 1987.

EDIT: And there is a book based on Arnold's own lectures:

MR2110624 Alekseev, V. B. Abel's theorem in problems and solutions. Based on the lectures of Professor V. I. Arnold. With a preface and an appendix by Arnold and an appendix by A. Khovanskii. Kluwer Academic Publishers, Dordrecht, 2004.


It is worth to read the first book on Group Theory -- W. Burnside, Theory of Groups of Finite Order -- to understand that abstract theory is usefull. By the way, sometime in the 19th century infinite groups were also considered as useless.

When I give lectures on the theory of groups for students, I end with Polya theory (a simplified version). In it the same group acts on different sets, and therefore it is difficult to manage without abstract groups.


I suggest to take a look at the first chapter of the book "Algebra. A graduate course" by M.I. Isaacs.


This may not be exactly what you wanted, since it assumes a little prior background in group theory, but Permutation Groups by Dixon and Mortimer certainly focuses on the perspective of permutation groups :).


Here is a very good free textbook: Group Theory, JS Milne: http://www.flooved.com/reader/3425

and from a physicist's point of view by Kundu, Calcutta: http://www.flooved.com/reader/3357#1

  • $\begingroup$ Those links appear to be broken. $\endgroup$ – Ben McKay Dec 21 '18 at 12:26

While not (?) really a book, the reader Group theory for Maths, Physics and Chemistry students by A. Cohen, R. Ushirobira and J. Draisma might offer some helpful ideas.


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