Size 5 is the smallest size > 1 that admits an Equation 677 magma (sizes 2, 3, 4 admit none — see their respective size pages), and there is exactly one such magma up to isomorphism: magma#e549b5f8.
Proof. Equation 677 forces every row of the Cayley table to be a permutation (the map z ↦ y ◇ z has inverse x ↦ x ◇ ((y ◇ x) ◇ y)), so candidates lie in the (5!)⁵ = 24,883,200,000 row-Latin tables. Backtracking that checks each instance of Eq677 as soon as the four rows it references are assigned visits only 6,409 nodes and returns 6 labeled magmas. Computing each one's canonical form (lex-min over all 5! = 120 relabellings) yields the same form for all six, so they are pairwise isomorphic.
Orbit–stabilizer then gives |Aut(M)| = 5! / 6 = 20: a transitive group of order 20 on 5 points is the Frobenius group AGL(1, 5) = F₅ ⋊ F₅* — the affine group on F₅, acting sharply 2-transitively.
[text written by Claude]
dwrensha · 2026-05-14 19:45:18
Size 5 is the smallest size > 1 that admits an Equation 677 magma (sizes 2, 3, 4 admit none — see their respective size pages), and there is exactly one such magma up to isomorphism: magma#e549b5f8492c9b6b5ad530e3aa4f39c6e23d08645ebd1b36a3c2de2a5a23bac5.
Proof. Equation 677 forces every row of the Cayley table to be a permutation (the map z ↦ y ◇ z has inverse x ↦ x ◇ ((y ◇ x) ◇ y)), so candidates lie in the (5!)⁵ = 24,883,200,000 row-Latin tables. Backtracking that checks each instance of Eq677 as soon as the four rows it references are assigned visits only 6,409 nodes and returns 6 labeled magmas. Computing each one's canonical form (lex-min over all 5! = 120 relabellings) yields the same form for all six, so they are pairwise isomorphic.
Orbit–stabilizer then gives |Aut(M)| = 5! / 6 = 20: a transitive group of order 20 on 5 points is the Frobenius group AGL(1, 5) = F₅ ⋊ F₅* — the affine group on F₅, acting sharply 2-transitively.
[text written by Claude]
dwrensha · 2026-05-14 18:24:20
Size 5 is the smallest size > 1 that admits an Equation 677 magma (sizes 2, 3, 4 admit none — see their respective size pages), and there is exactly one such magma up to isomorphism.
**Proof.** Equation 677 forces every row of the Cayley table to be a permutation (the map z ↦ y ◇ z has inverse x ↦ x ◇ ((y ◇ x) ◇ y)), so candidates lie in the (5!)⁵ = 24,883,200,000 row-Latin tables. Backtracking that checks each instance of Eq677 as soon as the four rows it references are assigned visits only 6,409 nodes and returns 6 labeled magmas. Computing each one's canonical form (lex-min over all 5! = 120 relabellings) yields the same form for all six, so they are pairwise isomorphic.
Orbit–stabilizer then gives |Aut(M)| = 5! / 6 = 20: a transitive group of order 20 on 5 points is the Frobenius group AGL(1, 5) = F₅ ⋊ F₅* — the affine group on F₅, acting sharply 2-transitively. So the unique class is the affine magma over F₅.
Canonical representative (lex-min relabelling):
```
0 2 1 4 3
3 1 4 0 2
4 3 2 1 0
2 4 0 3 1
1 0 3 2 4
```
[text written by Claude]
dwrensha · 2026-05-14 18:23:39
Size 5 is the smallest size that admits an Equation 677 magma, and there is exactly one such magma up to isomorphism.
**Proof.** Equation 677 forces every row of the Cayley table to be a permutation (the map z ↦ y ◇ z has inverse x ↦ x ◇ ((y ◇ x) ◇ y)), so candidates lie in the (5!)⁵ = 24,883,200,000 row-Latin tables. Backtracking that checks each instance of Eq677 as soon as the four rows it references are assigned visits only 6,409 nodes and returns 6 labeled magmas. Computing each one's canonical form (lex-min over all 5! = 120 relabellings) yields the same form for all six, so they are pairwise isomorphic.
Orbit–stabilizer then gives |Aut(M)| = 5! / 6 = 20: a transitive group of order 20 on 5 points is the Frobenius group AGL(1, 5) = F₅ ⋊ F₅* — the affine group on F₅, acting sharply 2-transitively. So the unique class is the affine magma over F₅.
Canonical representative (lex-min relabelling):
```
0 2 1 4 3
3 1 4 0 2
4 3 2 1 0
2 4 0 3 1
1 0 3 2 4
```
[text written by Claude]
dwrensha · 2026-05-14 19:46:43
dwrensha · 2026-05-14 19:45:18
dwrensha · 2026-05-14 18:24:20
dwrensha · 2026-05-14 18:23:39