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.]
Best Answer
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.