⟨1⟩ · index 9 · 9 colours
Groups, subgroups, colourings
Four things get called “the colouring”: a subgroup, a partition of the group into cosets, a set of orbits, and a picture. Three of them are the same object read three ways. The fourth — the picture — is not, and almost every argument about colour symmetry that goes in circles is circling that gap. This note fixes the dictionary, then walks the two smallest examples where every claim can be checked by hand, and ends with the lattice case.
The companion note, Are the subgroups of a group the same thing as its colourings?, proves the classification statement and counts the wallpaper cases. This one is the tutorial underneath it.
1. The dictionary
Let Γ be the symmetry group of an uncoloured figure — a wallpaper group, the symmetries of a torus grid, a finite rosette group. Colour the figure. The colouring is perfect when every symmetry in Γ permutes the colours: each g ∈ Γ takes all the blue to one single colour, all the red to one single colour, and so on. That gives a homomorphism ρ : Γ → Sym(colours).
Now fix one colour and follow four objects.
| Object | Definition | What it is in the picture |
|---|---|---|
| H | {g ∈ Γ : ρ(g) fixes the chosen colour} | the symmetries that leave that one colour where it is — they may move the others |
| k = [Γ : H] | the index | the number of colours, when Γ carries every colour to every other |
| the colours | the cosets of H | the colour of the motif you reach by applying g is the coset of g |
| K = core(H) | ⋂g gHg−1 = ker ρ | the symmetries that fix every colour — the colour-preserving group |
| Γ/K | the image of ρ | the group of colour permutations, the colour group |
A transitive perfect colouring with k colours is a subgroup of index k, and conversely. Not an analogy — the same data written twice.
Everything else in this note is about the three places where that sentence is easy to misapply: which cosets, orbits versus cosets, and the difference between a colouring and a pattern.
2. Which side? Left cosets, right cosets
Both answers are correct and they are not the same partition. What settles it is which side the group acts on, and that in turn is settled by how you compose two symmetries — a bookkeeping choice made before any mathematics happens.
Γ acting on the left
Write g·x for “apply the symmetry g to the point x”, so that (gh)·x = g·(h·x): the right-hand symmetry goes first. Then
- colours are the left cosets gH;
- the motif reached from the base motif by g has colour gH;
- g sends the colour xH to gxH.
Γ acting on the right
Write x·g, and read a product AB as “A then B” — the crystallographic habit, and the one this repository’s composition routines use. Then
- colours are the right cosets Hx;
- the motif reached from the base motif by g has colour Hg;
- g sends the colour Hx to Hxg.
They differ. In the symmetry group of a triangle, with H the two-element subgroup generated by one reflection s:
| Left cosets gH | {1, s} {r, rs} {r², r²s} |
|---|---|
| Right cosets Hg | {1, s} {r, r²s} {r², rs} |
Two of the three blocks are different sets. Pick one convention and hold it: mixing them gives a map that reverses products — ρ(gh) = ρ(h)ρ(g) — and then every relation you check comes out backwards. It is worth saying plainly that this site is not uniform about it: the interactive pages colour by left cosets under a left action, while the companion note works on the right. Neither is wrong; they are mirror translations of one statement.
The trap worth naming: the orbits of H acting on Γ by left multiplication are the right cosets, and vice versa. So “orbit” and “coset” sound interchangeable in conversation while pointing at opposite conventions.
3. The smallest complete example: the 3×3 torus
Take a 3×3 grid that wraps both ways, and let it act on itself by translation: A = ⟨X, Y | X³ = Y³ = 1, XY = YX⟩ = (ℤ/3)², of order 9. It has exactly six subgroups. Colour each cell by its coset; the ringed cell in each picture is the identity, and its block is H itself.
⟨Y⟩ · index 3 · 3 colours
⟨X⟩ · index 3 · 3 colours
⟨XY⟩ · index 3 · 3 colours
⟨XY²⟩ · index 3 · 3 colours
⟨X, Y⟩ · index 1 · 1 colour
Read off the dictionary. The number of colours is the index. The four subgroups of order 3 are four genuinely different colourings — ⟨X⟩ and ⟨Y⟩ stripe the torus two ways and ⟨XY⟩, ⟨XY²⟩ stripe it diagonally — even though all four are abstractly the same group ℤ/3. What distinguishes them is not what H is but how it sits in A.
The classification is of pairs (Γ, H), never of H alone. Two subgroups can be isomorphic as abstract groups, and even conjugate in some larger group, and still be different colourings.
A is abelian, so every subgroup is normal, H = K in every case, and every colour group is regular: the number of colours equals the order of the colour group. Nothing can go wrong here, which is exactly why it is the place to start. The 3×3 torus tutorial works the same six subgroups through three notations.
4. Cosets are not orbits
Now the smallest example where something does go wrong. Wrap a 2×2 grid and add the diagonal flip s. Mod 2 the quarter turn is the flip, so this group is Γ = (ℤ/2)² ⋊ ⟨s⟩ of order 8 — the dihedral group of the square, acting on four cells. It has ten subgroups, four of them not normal.
Here is the natural thing to do with a subgroup you want to draw: colour each orbit of H on the four cells. Ten subgroups, and only seven pictures come out:
⟨Y, X, XY⟩ = ⟨Ys, Xs, XY⟩ = Γ
one picture, 3 subgroups
⟨XY⟩ = ⟨s, XY, XYs⟩
one picture, 2 subgroups
⟨1⟩
order 1
⟨s⟩
order 2
⟨XYs⟩
order 2
⟨Y⟩
order 2
⟨X⟩
order 2
Three different subgroups — two of order 4 and the whole group — leave the picture blank, and two more share a picture. Colouring by orbits cannot name the subgroup that produced it, because a subgroup can be transitive on the cells without being everything.
Colouring by cosets always recovers H: H is the block containing the identity. Colouring by orbits of H on some other set need not — and the four cells are “some other set”, because each cell has its own symmetry.
That last clause is the whole of section 6, arriving early.
5. Regular, non-regular, and why H has to be named
A colouring is regular when H = K, which happens exactly when H is normal. Then the number of colours equals the order of the colour group Γ/K, and naming the colour group names the colouring.
When H is not normal the two part company. On the 2×2 torus take H = ⟨s⟩, of order 2: its core is trivial, so K = 1 and the colour group is all of Γ, order 8 — against four colours. Eight ≠ four, so “the colour group is Γ” does not say which colouring this is, and a symbol built from Γ and K alone cannot either. This is what the double slash in The Symmetries of Things is for, and why the subgroup has to be carried in the symbol next to the quotient.
Non-regular colourings, and why H must be named works this case out in full, including the two distinct subgroups that produce the same Conway symbol.
6. A colouring is not a pattern: where the motif sits
Everything so far colours the group, or a set the group permutes freely. A real pattern also has a motif, and the motif has symmetries of its own. Call S = {g ∈ Γ : g fixes the motif} — its stabiliser, the seat. The motifs of the pattern are then the cosets Γ/S, and the colouring is a map
Γ/S ⟶ Γ/H, gS ↦ gH,
which is well defined if and only if S ≤ H. The condition is not a technicality: it says that a symmetry which fixes the motif must fix the motif’s colour. Put a motif where a mirror runs through it and that mirror is forced into H — it cannot be one of the symmetries that swaps colours.
A colour group is a pair (Γ, H). A coloured pattern is a triple (Γ, H, S). The extra datum is where you put the motif, and it is the reason two pictures can have the same group and refuse to look alike.
This site’s own catalogue counts both. Over the seventeen wallpaper groups there are 46 two-colour groups and 88 two-colour pattern types; across all palette sizes up to six, 186 of the 286 colour groups carry more than one pattern, out of 609 types in all. Same group, different seat, different pattern — that is the ordinary case, not the exception.
The smallest instance needs no colour at all. The uncoloured group p2 carries two pattern types: pp7, with the motif in general position (S = 1), and pp8, with the motif seated on a 2-fold centre (S ≅ c2). One group, one colour, two patterns.
And the count that section 4 was really about: the colour classes of the motifs are the orbits of H on Γ/S, which are the double cosets H\Γ/S. When S = 1 those are just the cosets of H and the picture names H; when S ≠ 1 they need not, which is precisely why the orbit pictures collapsed.
7. Translations: sublattices, and what “horizontal and vertical” survives
Take Γ = p1 = ℤ², nothing but translations, and pick the two obvious generators — one horizontal, one vertical. Every element decomposes into those two. Does every subgroup?
No. In the given basis, the subgroups of index k are the lattices with a basis (a, 0), (c, d) where ad = k and 0 ≤ c < a, and only the ones with c = 0 are products aℤ × dℤ. At index 2 there are three subgroups and one of them is not a product:
ℤ × 2ℤ
a product of subgroups
2ℤ × ℤ
a product of subgroups
{(x, y) : x − y even}
the checkerboard — not a product
But now change basis. Smith normal form: for every sublattice Λ ≤ ℤ² there is some basis f₁, f₂ of ℤ² and integers d₁ | d₂ with Λ = d₁ℤf₁ ⊕ d₂ℤf₂. So the decomposition into two independent directions always exists — it just need not be the pair of directions you started with. The checkerboard is a product too, along its own diagonals.
That is the whole relation between the two questions, and it is why the counts differ:
| index k | sublattices of ℤ² | colour groups of p1 |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 3 | 1 |
| 3 | 4 | 1 |
| 4 | 7 | 2 |
| 5 | 6 | 1 |
| 6 | 12 | 1 |
| 7 | 8 | 1 |
| 8 | 15 | 2 |
The middle column is σ(k), the sum of the divisors — subgroups counted as sets. The right column is #{d : d² | k} — subgroups counted up to a change of basis, that is up to Aut(p1) = GL₂(ℤ). The three index-2 lattices above are one colour group drawn in three coordinate systems; the invariants that survive are exactly the Smith divisors (d₁, d₂), and ℤ²/Λ ≅ ℤ/d₁ × ℤ/d₂ is the colour group.
The picture behind the arithmetic is the torus. ℝ²/ℤ² is a torus; a sublattice Λ of index k gives a second torus ℝ²/Λ covering it k times over, and the k colours are the k sheets. The 3×3 grid of section 3 is this with Λ = 3ℤ²: nine cells, nine sheets, colour group (ℤ/3)². Sublattice, colouring, covering and finite quotient are four names for one object — which is why the torus is the right place to look when the wallpaper case gets confusing.
8. The short version
| Question | Answer |
|---|---|
| What is a colouring? | a subgroup H ≤ Γ, up to Aut(Γ) |
| What are the colours? | the cosets of H — left or right according to which side Γ acts on |
| Which symmetries keep the colours? | K = core(H); the colour group is Γ/K |
| When does the symbol determine the colouring? | when H = K (H normal, the regular case); otherwise H must be named too |
| Are orbits the same as cosets? | on Γ itself yes, on anything else no |
| Is a colouring a pattern? | no — a pattern also fixes the seat S ≤ H |
| Do sublattices decompose into two directions? | yes, but in their own basis, not necessarily yours |
Where to look next on this site: the pattern catalogue for the (Γ, H, S) triples with their seats drawn; the companion note for the classification theorem and the wallpaper counts; the designer for the same groups as animations, where a cyclic colour permutation is a shift in time.