# Reference Request: Group Theory via the Group Action Perspective

I am looking for a higher undergraduate or graduate level textbook that introduces group actions after groups just as many textbooks introduce modules after rings.

I think the semigroup/semigroup action vs. group/group action vs. ring/module ("ring action") symmetry is not currently exploited as much as it could be. Especially considering they are mild abstractions of a collection of plain vanilla functions acting on a set.

On the same note, a reference to an abstract algebra textbook with a significant chapter on semigroups (and actions) would be appreciated. A semigroup action is just a few points of data short of a finite automata. Because of this, and because of formal languages, semigroups show up a lot in computer science, and one can get quite far quite quickly. There are also deep algebraic results, see for example the Chomsky–Schützenberger enumeration theorem.

• I found Martin Isaac's book is very useful is this direction.... – Ripan Saha Aug 18 '15 at 20:25
• I think all the books define group actions after defining group :P – Bhaskar Vashishth Aug 18 '15 at 20:32
• What are you looking for exactly, and what direction. I feel like most books with group theory, or on group theory have group actions (the Sylow theorems for example). Plus are you focused on finite groups. Representation of group is all about groups acting on vector spaces. Geometric group theory is basically about groups acting on metric spaces. – Paul Plummer Aug 18 '15 at 20:36
• @BhaskarVashishth For an opposite direction, please see Arnold, V.I., ODE text. :) – Artem Aug 18 '15 at 20:44
• I'm looking for group actions appearing early in the exposition of groups, not in anything complicated. Many books prove many theorems in group theory, such as Sylow's, before group actions are introduced. The expositional order I'd like to see is: Functions on a finite set, semigroups and semigroup actions, bijections on a finite set, then groups and group actions. According to preference, one could interleave matrices acting on coordinate vectors. – ThoralfSkolem Aug 18 '15 at 20:45