Equation 677 Database

Steiner-line construction on the cyclic S(2, 5, 41) design. Size 41, fully idempotent, right-cancellative. Every pair of distinct elements lies in exactly one size-5 sub-magma, and there are 82 = 41*40/(5*4) such blocks, each point lying in 10 of them. Each block is isomorphic to magma#e549b5f8, the unique size-5 Eq 677 magma (the affine F_5 line). |Aut(M)| = 205 = 5 * 41, acting transitively on points. Aut has a regular Z_41 subgroup of translations, and the point-stabilizer is the Z_5 subgroup of F_41* generated by the order-5 element zeta = 10 (so {1, 10, 18, 16, 37} acts by multiplication). The structure is the Sharply 1-transitive piece of AGL(1, 41) restricted to a Z_5 multiplier group; equivalently Aut ~ Z_41 : Z_5. In the suggested reorder, elements are labeled by F_41 = Z/41 in an order where translation x -> x+1 is an automorphism. The operation then takes the translation-invariant form x*y = x + f(y - x) mod 41 with f a non-linear permutation of F_41. Concretely f(y) = alpha(y) * y where alpha(y) takes only 4 values - the 4 roots of the cyclotomic polynomial Phi_10 in F_41: Phi_10 roots = {4, 23, 25, 31} (these are the primitive 10th roots of unity in F_41*) alpha is constant on each of 4 size-10 subsets that partition F_41*. Each size-10 "slope class" is a union of two Z_5-cosets in F_41*/<zeta>: alpha=31: y in {1, 2, 10, 16, 18, 20, 32, 33, 36, 37} alpha=4: y in {4, 5, 8, 9, 21, 23, 25, 31, 39, 40} alpha=25: y in {3, 6, 7, 13, 14, 17, 19, 26, 29, 30} alpha=23: y in {11, 12, 15, 22, 24, 27, 28, 34, 35, 38} This is a "Tao Type II piecewise" recipe: f is not multiplication by a single scalar, but each Z_5-coset gets its own primitive-10th-root slope. The four slopes are Galois-conjugate, and the assignment of slopes to cosets is what makes this Eq 677. Block structure: the 82 blocks form 2 Aut-orbits (each of size 41), the orbits being the translate-classes of two base blocks under Z_41 (the standard difference family for a cyclic (41, 5, 1)-BIBD). The 10 blocks through 0 split as 2 Z_5-orbits of 5 blocks each. Compares with the F_p affine line magmas (size p, p ≡ 1 mod 10, e.g. p = 11, 31, 61) which are translation-invariant but with f(y) = alpha*y a single linear map; those are also Steiner-line magmas, but their S(2, 5, p) is "linear" - this size-41 magma uses a more general Tao Type II f and gives a (likely) inequivalent design. [text written by Claude]