Recipe

List every element of D_n

Use when: you're given a regular n-gon, or just its symmetry count 2n, and need every element of D_n written with the r^k / r^k s labels

Goal: a complete list of D_n's 2n elements, split into n rotations and n reflections

Taught in The dihedral group Dₙ

1. 1

   Read the polygon's number of sides (or vertices) off as n; the group's order is always 2n, twice the subscript, never n itself.

   Example

   A regular hexagon has n=6, so |D_6|=2(6)=12.

   • n ≥ 3 — an ordinary polygon's symmetry group; continue to Step 2 as written.
   • n = 1 or n = 2 — the degenerate case; the order is still 2n, but (unlike n≥3) the group turns out to be abelian.

   ⚡ Shortcut

   If only the order is needed, not every individual labeled element, stop right here: |D_n|=2n already is the whole answer.

   Why: every labeling scheme below depends on knowing n first, and getting the order right up front catches the common slip of reading the subscript itself as the element count.

2. 2

   Write e, r, r^2, \ldots, r^{n-1}: n total powers of the chosen rotation generator r, stopping just before r^n=e repeats the list.

   Example

   For D_3: e, r, r^2.

   Why: r has order n by construction, since n rotations by 360°/n each bring the polygon back to its starting position after exactly n steps.

3. 3

   Write s, rs, r^2s, \ldots, r^{n-1}s: multiply the fixed reflection s by each rotation power in turn.

   Example

   For D_3: s, rs, r^2s.

   Why: section 3 proves every reflection of the n-gon takes exactly this form, so this list is guaranteed to catch all of them, none missed or repeated.

4. 4

   Union the n rotations and n reflections into one list of 2n distinct elements, then check that count against 2n.

   Example

   For D_3: \{e, r, r^2, s, rs, r^2s\}, six elements, matching |D_3|=6.

   Why: matching 2n confirms no label was written twice and none of the n-gon's actual symmetries was left out.