A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.
The dihedral group Dₙ is the group of symmetries of a regular n-gon, including both rotations and reflections. It has order 2n and is generated by a rotation r and a reflection s with relations rⁿ = e, s² = e, and srs = r⁻¹. Dihedral groups are prototypical non-abelian groups.
Start with a concrete object: an equilateral triangle. Pick it up, and ask: what actions can you perform on it that leave it looking exactly the same as when you started? You can rotate it by 120° or 240°, or you can flip it across any of three axes of symmetry. Together with doing nothing (the identity), these six actions form a group — the dihedral group D₃, the symmetry group of a regular triangle. The same idea generalizes: Dₙ captures all rotations and reflections of a regular n-gon, giving 2n elements in total (n rotations and n reflections).
The group is built from just two generators. The rotation r advances the polygon by one step: rotating by 360°/n. Applying r repeatedly gives r, r², r³, ..., rⁿ = e (back to start). The reflection s flips the polygon across one fixed axis. Applying s twice returns you to the start: s² = e. So far, this looks like two cyclic groups. The critical relation that ties them together — and makes Dₙ non-abelian — is srs⁻¹ = r⁻¹, or equivalently srs = r⁻¹. This says: if you flip, then rotate, then flip back, you get the reverse rotation. In practical terms: rotation and reflection do not commute. Doing a flip then a rotation gives a different result than doing the rotation then the flip.
Your prerequisite on permutations gives you the tools to make this concrete. The n vertices of the polygon can be labeled 1 through n. Every symmetry — rotation or reflection — permutes those labels. For D₃, the six symmetries correspond to six permutations of {1, 2, 3}, and you can verify the non-commutativity by composing specific permutations and watching the order matter. The sign of those permutations (even or odd) is also meaningful: the n rotations are all even permutations, while the n reflections are all odd. This connects Dₙ to the alternating group Aₙ as a subgroup of the full symmetric group Sₙ.
Dihedral groups occupy an important place in abstract algebra for two reasons. First, they are among the smallest examples of non-abelian groups, making them ideal for developing intuition about non-commutativity. D₃ is actually isomorphic to S₃, the symmetric group on 3 elements — the smallest non-abelian group. Second, the relation srs = r⁻¹ is a presentation by generators and relations, a technique that generalizes to defining many other groups. Understanding how Dₙ is presented this way prepares you to recognize the same structure pattern in group actions and in the theory of normal subgroups: the subgroup of rotations ⟨r⟩ is always normal in Dₙ (since srs⁻¹ = r⁻¹ ∈ ⟨r⟩), while the reflection subgroups are generally not normal.