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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
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.
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.
The four readings and six extra questions were written for this dataset. The flashcards include material from the imported fact collection. The linked TED-Ed talk is by Michael Staff.
More to read
Why one destination corresponds to one coset
Choose a point x in a group action. Two elements g and h send it to the same point exactly when h^-1g fixes x. This is also the condition for g and h to belong to the same left coset of G_x.
The map gG_x -> g.x therefore gives one orbit point for each stabiliser coset. It is well defined, reaches every orbit point and assigns different points to different cosets. For finite G, each coset contains |G_x| elements. Multiplying the number of cosets by that size gives |G|.
This argument does not require the stabiliser to be normal. The cosets form a set for counting; no quotient-group multiplication is needed. It also does not require different group elements to act differently.
From conjugation to the centre
Conjugation sends x to gxg^-1. An element g fixes x in this action exactly when gx = xg. Thus the stabiliser is the centraliser C_G(x), and the orbit is the conjugacy class of x.
The centraliser concerns commuting with one specified element. The centre concerns commuting with every element. An element is central exactly when its conjugacy class is a singleton.
For finite G, the conjugacy classes partition G. Add the singleton classes together as |Z(G)| and add each other class size once to obtain the class equation. If |G| is a positive power of a prime p, each noncentral class size is divisible by p. The centre size is then divisible by p too. It is positive because it includes the identity, so it is at least p.
The action includes its set
Specifying a group does not specify an action. You must also name the set and say how each group element moves its points. Cube rotations can act on faces or vertices. The group has 24 elements in either case, but the point stabilisers have different sizes.
An action may have several orbits. A transformation swapping two points and fixing a third gives one two-point orbit and one singleton orbit. Use the orbit of the selected point in orbit-stabiliser, not automatically the whole acted-on set.
An action may also be nonfaithful. Let additive Z4 act on Z2 by addition modulo 2. Elements 0 and 2 fix both points; 1 and 3 swap them. Four group elements give only two permutations. Each point still has an orbit of size two and a stabiliser of size two.
Specify the moves before comparing cube groups
A rotation of a solid cube moves all its points together. Its rotational symmetry group has 24 elements. Preserving a face as a set does not require leaving each point of that face in place.
A puzzle-layer turn moves some pieces relative to others. To define a puzzle-move group, specify the allowed moves and the positions being permuted. Also say whether turning the whole puzzle changes the state or only changes the observer's viewpoint.
These choices affect comparisons with the solid cube's rotation group. For example, a face-turn model keeping the six centre positions fixed does not automatically include whole-cube rotations that exchange those positions. A model allowing whole-puzzle rotations has different allowed transformations. The 24 solid-cube rotations are therefore not, without a stated convention, a subgroup of every puzzle-move group.
Explore more
2 · Watch
Take it off screen
Where it sits
14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.