--- title: "Group Actions, Orbits and the Class Equation" description: "A group acting on a set splits it into orbits, and each orbit has the size of an index of a stabiliser. Counting one set two ways with that relation proves results no direct argument reaches." canonical: https://lightmysky.com/learn/mathematics/group-actions-orbits-and-the-class-equation-mt_urut0TFTOR source: https://lightmysky.com/learn/mathematics/group-actions-orbits-and-the-class-equation-mt_urut0TFTOR.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Group Actions, Orbits and the Class Equation A group acting on a set splits it into orbits, and each orbit has the size of an index of a stabiliser. Counting one set two ways with that relation proves results no direct argument reaches. Subject: Mathematics · Area: Abstract Algebra · Ages 22 to 23 Page: https://lightmysky.com/learn/mathematics/group-actions-orbits-and-the-class-equation-mt_urut0TFTOR ## Ready when they can - Identify the orbits and stabilisers of a given action - Apply the orbit-stabiliser relation to a symmetry group - Use the class equation to show that a group of prime-power order has a nontrivial centre ## Lesson: Count symmetries with orbits When a group acts on a set, you let each group element shuffle the set while respecting the group law. The orbit of a point is the set of places it can travel to, and the stabiliser is the subgroup of elements that fix it. The orbit stabiliser relation says orbit size times stabiliser size equals the group order. **Example.** Rotations of a cube send any face to any of the 6 faces, and 4 quarter turns fix a chosen face, so the rotation group holds 6 times 4, or 24, elements. Counting vertices agrees: 8 vertices times 3 spins about a body diagonal. A tetrahedron has 4 faces with 3 spins fixing each, giving 12 symmetries. You can sort a group by letting it act on itself by conjugation, which turns orbits into conjugacy classes. Counting the set two ways gives the class equation: the group order equals the center size plus the larger class sizes. In a group of prime power order each larger class size shares the prime factor, so the center holds more than the identity, and a group of 27 elements carries central elements beyond the identity. **Tip.** When the action is transitive you face a single orbit, so division does the work. A group of order 12 with a point stabiliser of size 3 has orbits of 12 divided by 3, or 4, points. A group of order 20 acting transitively on 5 objects has stabilisers of 20 divided by 5, or 4, elements. **Recap.** Orbits and stabilisers split the group order between them, and the class equation turns that split into facts like nontrivial centers. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Groups, Subgroups and Symmetry](https://lightmysky.com/learn/mathematics/groups-subgroups-and-symmetry-mt_HFAhcaGo2L) - [Normal Subgroups and Quotient Groups](https://lightmysky.com/learn/mathematics/normal-subgroups-and-quotient-groups-mt_RU7A-t7wQ5) ## Opens up - [Noether's Theorem: Symmetry and Conserved Quantities](https://lightmysky.com/learn/science/noethers-theorem-symmetry-and-conserved-quantities-mt_kAoOn8syBK) - [Characters and the Orthogonality Relations](https://lightmysky.com/learn/mathematics/characters-and-the-orthogonality-relations-mt_RleR7YGXjX) - [The Sylow Theorems and Groups of Small Order](https://lightmysky.com/learn/mathematics/the-sylow-theorems-and-groups-of-small-order-mt_vg2VYkLuz8) - [Group Representations and the Group Algebra](https://lightmysky.com/learn/mathematics/group-representations-and-the-group-algebra-mt_y5JlvQGKYW)