Equation 677 Database

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]