--- title: "Group Actions, Orbits and the Class Equation" description: "A group action partitions a set into orbits. An orbit corresponds to the left cosets of its point stabiliser. For finite groups, this correspondence gives orbit-stabiliser counting; conjugation gives " 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: 2026-11-12 --- > **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 action partitions a set into orbits. An orbit corresponds to the left cosets of its point stabiliser. For finite groups, this correspondence gives orbit-stabiliser counting; conjugation gives the class equation. 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 A group action assigns each element of G a permutation of a set X. Write g.x for the resulting point. The rules are e.x = x and (gh).x = g.(h.x). The orbit of x is {g.x: g in G}. Its stabiliser G_x is {g: g.x = x}, a subgroup of G. Orbits contain points; stabilisers contain group elements. Fixing one point does not require fixing the others. Recall that left cosets partition a group. The map gG_x -> g.x is a bijection from stabiliser cosets to orbit points. For finite G, |orbit of x| times |G_x| = |G|. Divide by stabiliser size to find orbit size, or by orbit size to find stabiliser size. A transitive action has one orbit containing the whole set; otherwise use the formula separately for each orbit. *(drawing: Each destination accounts for one coset of fixing elements.)* **Example.** For a transitive symmetry action, multiply orbit size by stabiliser size. A regular pentagon's planar symmetries act on its 5 vertices, with 2 symmetries fixing a vertex, so the group has 5 times 2 = 10 elements. Cube rotations act on all 6 faces. A face's setwise stabiliser consists of turns of 0, 90, 180 and 270 degrees about the axis through opposite face centres. These turns can move points within the face. A cube-preserving rotation fixing a vertex must permute its three incident edges while preserving cyclic order. The allowed turns about its body diagonal are multiples of one third of a full turn. Cube rotations are transitive on 8 vertices. A regular tetrahedron's 4 faces form one orbit, with 3 rotations preserving a chosen face as a set. To count a finite group, multiply orbit size by stabiliser size. *(drawing: For the regular pentagon's planar symmetry group, five orbit points each account for two symmetries.)* To list orbits, apply every group element to a starting point. Repeat at a point not yet listed. Distinct orbits are disjoint. A point fixed by every group element has a singleton orbit. To list a stabiliser, retain exactly the group elements returning the chosen point to itself. In a permutation action, an element fixing 1 may still exchange other letters. **Example.** Different group elements may give the same permutation. Additive Z8 acts on Z4 by a.x = a + x modulo 4. Both 0 and 4 fix every residue. The orbit of 0 contains all four residues, and its stabiliser is {0, 4}. Thus 4 times 2 = 8. This action is nonfaithful: it does not distinguish every group element. Conjugation is the action g.x = gxg^-1 on G itself. Its orbits are conjugacy classes. Its stabiliser at x is the centraliser C_G(x), consisting of elements commuting with x. For finite G, |class of x| = |G| / |C_G(x)|. Thus each class size divides |G|. Exactly |G| - |C_G(x)| elements do not commute with x. A singleton class means x belongs to the centre Z(G). Conjugacy classes partition a finite group. The class equation is |G| = |Z(G)| + the sum of noncentral class sizes, counting each class once. Central elements give singleton classes. To find the centre size, subtract all noncentral class sizes from the group order. Suppose |G| = p^n, with p prime and n at least 1. Each noncentral class size is greater than 1 and divides p^n. It is therefore a positive power of p and divisible by p. The class equation makes |Z(G)| divisible by p too. The identity is central, so |Z(G)| is positive. Hence |Z(G)| is at least p and the centre is nontrivial. **Recap.** Orbit size multiplied by stabiliser size gives the group order. The class equation uses this relation to prove results such as nontrivial centres. *The lesson, questions, worked solutions and base flashcards were written for this dataset.* ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Normal Subgroups and Quotient Groups](https://lightmysky.com/learn/mathematics/normal-subgroups-and-quotient-groups-mt_RU7A-t7wQ5) ## Optional connections - [Groups, Subgroups and Symmetry](https://lightmysky.com/learn/mathematics/groups-subgroups-and-symmetry-mt_HFAhcaGo2L) ## 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)