Group Actions (expository notes)
by Keith Conrad · University of Connecticut
The most thorough free text devoted entirely to group actions: Keith Conrad's long expository paper on group actions. It covers definitions, faithful and free actions, orbits and stabilizers, the orbit-stabilizer theorem, Burnside's lemma, p-group congruences, and new proofs of Cauchy's theorem, closing with applications to spacetime symmetry.
More resources on Group Actions
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
Group Properties Wiki
The Groupprops wiki entry on group actions, giving equivalent definitions, related notions such as orbits, stabilizers, transitive and faithful actions, and links to many worked examples and facts. Readers can use it as a precise reference while studying group theory.
Abstract Algebra
Learn group actions with Prof. Benedict Gross! This abstract algebra course explores key concepts with clear explanations.
Chapter 2: Orbit-Stabiliser Theorem | Essence of Group Theory
A roughly 12-minute animated explainer that derives the orbit-stabiliser theorem by counting the symmetries of a regular octagon, defines orbits and stabilisers intuitively, and gives a general proof for finite groups linked to Lagrange's theorem.
Visual Group Theory, Lecture 5.1: Groups acting on sets
A 32-minute lecture that defines group actions through action graphs and fixed-point tables, using the symmetries of the square as a running example. It opens a seven-part unit covering orbit-stabilizer, fixed points, Cauchy's theorem, p-groups and Sylow, with free slides. University-level lecture specifically on group actions, visual and concrete: the best free video bridge before Conrad's notes.
Abstract Algebra: Theory and Applications, Chapter 14: Group Actions
The gentlest rigorous entry point; its Burnside section is built on concrete counting problems, fitting the topic's combinatorics goal. Free open-textbook chapter covering groups acting on sets, the class equation, and Burnside's Counting Theorem. Worked examples include colorings of a square and switching functions, followed by reading questions, exercises and a programming exercise.