Equation 677 Database

Carrier: Z5 x Z5, element (a,b) encoded as 5*a + b. Operation: x @ y = x + f(y - x), where f is the permutation of Z5 x Z5 given, in the encoding above, by f = [0, 4, 3, 2, 1, 11, 13, 19, 23, 15, 24, 5, 17, 6, 22, 9, 20, 12, 21, 7, 16, 18, 14, 8, 10] so the translations of Z5 x Z5 act as automorphisms, simply transitively. f is NOT additive up to its constant, so this is not a linear model: it is outside the affine family x @ y = a*x + b*y + c. Construction: melvyn lab session 66. E677 magmas admitting a simply transitive automorphism group are exactly the pairs (L, f) with L a finite group and f a permutation of it, via x @ y = x * f(x^-1 y); substituting into the master equation collapses the n^2 instances of E677 to n equations in f alone, f(u) * f(f(u)^-1) = u * g(u^-1 g(u)) for all u, g = f^-1. Searching that equation over the two groups of order 25 is exhaustive and cheap, and it returns five isomorphism classes at this size. Four are already here: magma#d732efd172ca4d7b (the non-quasigroup), magma#053cceeb7cc1, magma#05017728ce1c and magma#09d21ec32e46. This is the fifth. Invariants: order 25; satisfies E255 at all 25 points; quasigroup; all 25 elements idempotent; diagonal orbit length l = 1 everywhere; no right identity; |Aut| = 100; Fix(f) = {0} and the displacement D(s) = f(s) - s is a bijection, which is what makes it right-cancellative. Verification: E677 y@(x@((y@x)@y)) = x at all 625 pairs, and independently the master equation (y@x)@y = L_x^{-1}(L_y^{-1}(x)) at all 625 pairs; rows permutations; rows pairwise distinct; E255 at all 25 points. Two implementations sharing no code agree.