Where the double slash comes from
Non-regular colourings, and why H must be named
The 2×2 torus group has four non-normal subgroups. Each defines a colouring with four colours but a colour group of order eight — non-regular. Two of them produce the same Conway symbol and yet are different colourings.
Regular versus non-regular
A colouring is regular when the number of colours equals the order of the colour
group, which happens exactly when H = K. Compare:
H = K = {e, XY, s, XYs}, a
normal subgroup of index 2. Two colours, colour group C₂ of
order 2. Colours = group order, so H is recoverable from K and
Conway writes a single slash.H has order 2 and
K = core(H) = {e}. Four colours, colour group of order 8.
Eight ≠ four, so H is not recoverable — double slash. The picture
is trivial; all the structure is in the action.The action is where the content is
For the rainbow pattern every element of Γ permutes the four colours. Rows shaded
green lie in H (they fix colour A); the darker row is
K.
| element | order | colour permutation | fixes A | in K |
|---|---|---|---|---|
e | 1 | identity | ✓ | ✓ |
X | 2 | (A C) (B D) | ||
XY | 2 | (A D) (B C) | ||
XYs | 2 | (A D) | ||
Y | 2 | (A B) (C D) | ||
s | 2 | (B C) | ✓ | |
Xs | 4 | (A C D B) | ||
Ys | 4 | (A B D C) |
Only the identity fixes every colour, so K = {e}. Two elements fix
A, so H has order 2 and index 4 — the four colours. The map
Γ → S₄ is injective but lands in a subgroup of index 3, which is exactly
the gap between 8 and 4.
The two cases
H = {e, s}
This one is realized by the grid itself: the four cells are the coset space Γ/⟨s⟩, and the rainbow pattern is its colouring.
Each thick coloured edge is one coset of H —
one colour. Thin grey edges change the colour.
| A | {e, s} | sends A → A |
| B | {Y, Ys} | sends A → B |
| C | {X, Xs} | sends A → C |
| D | {XY, XYs} | sends A → D |
| A | {e, s} |
| B | {XYs, e} |
| C | {XYs, e} |
| D | {e, s} |
2 distinct stabilizers for 4 colours, since |N(H)|/|H| = 4 / 2.
G = Γ (8) ·
H = order 2, index 4 ·
K = core(H) = {e}
· 4 colours, colour group of order 8
→ non-regular
H = {X, e}
This one is not realized by the cells: ⟨X⟩ has only two orbits on them, so the colouring needs a different four-element Γ-set. Drawn abstractly, as the four cosets.
Each thick coloured edge is one coset of H —
one colour. Thin grey edges change the colour.
| A | {X, e} | sends A → C |
| B | {Ys, s} | sends A → B |
| C | {XY, Y} | sends A → D |
| D | {XYs, Xs} | sends A → D |
| A | {e, s} |
| B | {XYs, e} |
| C | {XYs, e} |
| D | {e, s} |
2 distinct stabilizers for 4 colours, since |N(H)|/|H| = 4 / 2.
G = Γ (8) ·
H = order 2, index 4 ·
K = core(H) = {e}
· 4 colours, colour group of order 8
→ non-regular
Why the symbol is not enough
Both cases give identical G, identical K, and the same number of
colours — therefore the same Conway symbol Γ⁴//{e}:
H = ⟨s⟩ |
H = ⟨X⟩ | |
|---|---|---|
G | Γ, order 8 | Γ, order 8 |
K = core(H) | {e} | {e} |
| colours | 4 | 4 |
| symbol | Γ⁴//{e} | Γ⁴//{e} |
| realized on the cells? | yes | no |
| the colourings are | different | |
They differ because ⟨s⟩ and ⟨X⟩ are
not conjugate — they lie in the two separate classes of non-normal
subgroups — so the Γ-sets Γ/⟨s⟩ and
Γ/⟨X⟩ are not isomorphic. Same symbol, two colourings.
That is the whole reason Conway's notation carries three groups rather than two. The double
slash in Gⁿ//K is a warning that H has been
suppressed and cannot be reconstructed; the full form Gⁿ/H/K names it.
On an abelian ambient this never arises — every subgroup is normal, H = K
always, and a single slash suffices.
Generated by scripts/generate_torus_nonregular.py. Companion to the Cayley graph page and the torus tutorial.