Equation 677 Database

Z_4-symmetric Steiner-line magma at size 65 (the unique multi-stabilizer variant). Size 65 = 5 * 13, fully idempotent, RC. 208 size-5 sub-magmas on a cyclic S(2, 5, 65) Steiner system (each = the F_5 affine line magma#e549b5f8). Every pair of distinct elements lies in exactly one block. |Aut(M)| = 260 = 4 * 65. Aut is Z_65 : Z_4: the regular Z_65 = Z_5 x Z_13 translation, and the point-stabilizer Z_4 = {1, 34, 51, 44} = <34> is an order-4 cyclic subgroup of Z_65*. Note: 34^2 = 51, 34^3 = 44, 34^4 = 1 mod 65; 51 = -14 mod 65 is the unique involution in this Z_4. The stabilizer acts by multiplication. In the suggested reorder, Z_65 labeling makes translation x -> x+1 an automorphism. The operation x*y = x + f(y - x) mod 65 with f a non-linear permutation of Z_65. The Z_4 multiplicative stabilizer forces f to be Z_4-equivariant: f(34y) = 34 * f(y), so the slope function alpha(y) = f(y)/y (where defined) is constant on Z_4-orbits in Z_65*. Slope analysis on Z_65* (the 48 units mod 65): alpha takes 4 distinct values, each on a size-12 Z_4-coset: alpha=3: on {17, 22, 28, 33, 38, 42, 47, 53, 57, 58, 62, 63} (coset of 17 mod Z_4) alpha=22: on {1, 11, 19, 29, 34, 41, 44, 49, 51, 56, 59, 61} (coset of 1, includes Z_4 itself) alpha=35: on {2, 3, 7, 8, 12, 18, 23, 27, 32, 37, 43, 48} (coset of 2) alpha=51: on {4, 6, 9, 14, 16, 21, 24, 31, 36, 46, 54, 64} (coset of 4) The 4 slope values {3, 22, 35, 51} are partly Z_65* elements (3, 22, 51) and partly not (35 is a multiple of 5, so non-unit mod 65 - a phenomenon unique to composite sizes). This is the analog of magma#b7e8bf90 (size 41, Z_5-symmetric) and magma#0bcf3cca (size 61, Z_15-symmetric): the "most symmetric" Steiner-line magma at its size, with the largest available cyclic multiplicative stabilizer matching the (F_q*)^{|Phi-roots|}-structure. Note 65 has no F_q (since composite) and no Phi_10 roots in Z_65, so the analog uses a Z_4 multiplicative stabilizer instead of a multi-root-based Z_5 or Z_15. Among the 9 fully idempotent size-65 Steiner-line magmas, this is the unique one with |Aut| > 65. The other 8 have only Z_65 translation symmetry (|Aut| = 65) and split by slope-class-size patterns into 3 sub-types (see size-65 page commentary). [text written by Claude]