ISEGORIA / MATH ENCYCLOPEDIA
Group theory: symmetry as algebra
Compose symmetries, inspect group tables, and count orbits.
Before you begin: Functions and composition
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. Order matters
Number the vertices of a regular polygon and attach a small flag, then apply the same two symmetries in both orders: reflect in the vertical mirror and rotate anticlockwise by k steps. The outline never changes, but the labels and the flag show which symmetry you ended with. Press Play to watch both compositions move.
Worked example. For a square with k = 1, reflect then rotate gives rs, while rotate then reflect gives r³s: the labels end up in different places.
Watch out. An unchanged outline does not mean every vertex stayed fixed.
When do the two displayed compositions agree?
When 2k is divisible by n, including zero rotation and a half-turn for even n.
2. A multiplication table
A cell in row a and column b records the product a·b. For the rotation group Cₙ this is addition of steps modulo n. Switch to the full dihedral group Dₙ to add the reflections: the table stops being symmetric about its diagonal, because rotations and reflections do not commute. Click any cell to choose a and b, and find the inverse of a where the identity appears in its row.
Worked example. In C₅, 3+4 ≡ 2 modulo 5, and the inverse of 3 is 2. In D₄, r·s = rs but s·r = r³s.
Watch out. This rotation subgroup is commutative; the full dihedral group generally is not.
Why does each row contain every group element once?
Adding a fixed element is a bijection, with subtraction as its inverse.
3. Orbits, stabilizers, and cosets
A subgroup of symmetries acts on the vertices. Generate it from the rotation by k steps, optionally together with the reflection s, and click a vertex x. Its orbit is highlighted, and on the right the group elements are sorted by where they send x: each column is a coset of the stabilizer Gₓ, so every orbit point is hit by exactly |Gₓ| elements.
Worked example. For n=6 and k=2, the orbits are {0,2,4} and {1,3,5}.
Watch out. For the rotation subgroup every stabilizer is trivial; once the reflection is included, a vertex on the mirror line is fixed by s, and orbits can have different sizes.
How many orbits does the rotation subgroup ⟨rᵏ⟩ have?
gcd(n, k): each orbit has n/gcd(n, k) vertices, the order of the subgroup.