Group Actions, Orbits and Stabilisers
Handbook guide to group actions, orbits and stabilisers with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
Groups Acting on Sets
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
The Orbit-Stabilizer Theorem
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Core definitions and structural results
The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.
Cayley’s Theorem Every group is isomorphic to a group of permutations.
Cayley’s Theorem Every group is isomorphic to a group of permutations.
Proof / verification strategy: Construct the natural group homomorphism, identify its kernel and image, and then use quotient structure to prove the claimed isomorphism or correspondence.
Definition
The group G acts on the set X if for each g ∈G there is a mapping x →gx of X into itself, such that (1) h(gx) = (hg)x for every g, h ∈G and x ∈X and (2) 1x = x for every x ∈X. As in (5.1.1), x →gx defines a permutation of X. The main point is that the action of g is a permutation because it has an inverse, namely the action of g−1. (Explicitly, the inverse of x →gx is y →g−1y.) Again as in (5.1.1), the map from g to its associated permutation Φ(g) is a homomorphism of G into the group SX of permutations of X.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Examples
1. (The regular action) Every group acts on itself by multiplication on the left, as in (5.1.1). In this case, the homomorphism Φ is injective, and call the action is faithful. [Similarly, one can define an action on the right by (xg)h = x(gh), x1 = x, and then G acts on itself by right multiplication. The problem is that Φ(gh) = Φ(h) ◦Φ(g), an antihomomorphism.
Proof / verification strategy: Treat this as an illustration of the surrounding definitions. Recompute the stated relations directly and keep the numerical or structural choices local to the example.
Definition
Suppose that the group G acts on the set X. If we start with the element x ∈X and successively apply group elements in all possible ways, we get B(x) = {gx : g ∈G} which is called the orbit of x under the action of G. The action is transitive (we also say that G acts transitively on X) if there is only one orbit, in other words, for any x, y ∈X, there exists g ∈G such that gx = y. Note that the orbits partition X, because they are the equivalence classes of the equivalence relation given by y ∼x iffy = gx for some g ∈G.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Examples
1. The regular action of G on G is transitive, and the stabilizer of x is the subgroup {1}. 2. The trivial action is not transitive (except in trivial cases), in fact, B(x) = {x} for every x. The stabilizer of x is the entire group G. 3.
Proof / verification strategy: Treat this as an illustration of the surrounding definitions. Recompute the stated relations directly and keep the numerical or structural choices local to the example.
The Orbit-Stabilizer Theorem
The Orbit-Stabilizer Theorem Suppose that a group G acts on a set X. Let B(x) be the orbit of x ∈X, and let G(x) be the stabilizer of x. Then the size of the orbit is the index of the stabilizer, that is, |B(x)| = [G : G(x)]. Thus if G is finite, then |B(x)| = |G|/|G(x)|; in particular, the orbit size divides the order of the group.
Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.
Proposition
If H and K are subgroups of the finite group G, then |HK| = |H||K| |H K|.
Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.
Quick-reference relationships
Problem-solving workflow
Identify the ambient group
State the operation, identity, inverses and whether commutativity is available.
Locate the relevant subgroup structure
Check generated subgroups, normality, cosets, stabilisers or direct factors before applying a theorem.
Use the correct counting or mapping tool
Choose coset counting, orbit counting, a homomorphism, a quotient or a group action as appropriate.
Check hypotheses explicitly
Finite-order assumptions, normality, prime-power divisibility and coprimality conditions are not interchangeable.
Translate the result back to structure
Interpret the result as a statement about subgroup size, quotient structure, action, decomposition or presentation.
Verify with a small model
Use a cyclic, symmetric, dihedral or modular example to test the logic without treating the example as the theorem.
Worked-solution emphasis from the supplied source
The supplied worked solutions for this section repeatedly test order, module, homomorphism, exact, injective, factor, norm, Tor. These checks are used here as verification themes rather than copied as answer text.
The supplied worked solutions for this section repeatedly test order, subgroup, coset, kernel, homomorphism, exact, injective, orbit. These checks are used here as verification themes rather than copied as answer text.
Common mistakes and boundary conditions
- Treating left and right cosets as identical without normality.
- Assuming the converse of a subgroup-order divisibility result.
- Confusing the order of a group with the order of one of its elements.
- Using quotient multiplication before checking that the subgroup is normal.
Verification checklist
- State the ambient algebraic structure and operation before applying a theorem.
- Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
- Distinguish a definition from a theorem that follows from it.
- Check whether a map is well-defined before using its kernel, image, inverse or induced map.
- Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
- When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.
Source coverage map
| Source section | Subject | PDF pages analysed |
|---|---|---|
| 5.1 | Groups Acting on Sets | 85–85 |
| 5.2 | The Orbit-Stabilizer Theorem | 86–88 |
Related Mathematics pages
Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.