Equation 677 Database

Size-21 = 3·7 idempotent right-cancellative magma satisfying Eq 677 and Eq 255, in the Steiner-S(2, 5, 21) = PG(2, 4) family. Every pair of distinct elements generates a 5-element sub-magma (≅ unique size-5 Eq 677 magma magma#e549b5f8); the 21 sub-magmas form the lines of the projective plane PG(2, 4), with each point lying on 5 lines. Distinguishing feature: |Aut(M)| = 21 acting regularly on the 21 elements, with element orders {1, 3, 7} but NO order-21 element. So Aut(M) is the **non-abelian Frobenius group F_21 = Z_7 ⋊ Z_3** — NOT the cyclic Z_21 group. The 21 automorphisms break down as: 1 identity, 14 of order 3 (the Z_3 generators × Z_7 representatives), and 6 of order 7 (the Z_7-translations). Contrast with the third size-21 PG(2, 4) magma magma#b1cfacfa, which has |Aut| = Z_21 cyclic (with order-21 elements present). The two cases correspond to the two non-isomorphic groups of order 21: Z_21 abelian vs. F_21 = Z_7 ⋊ Z_3 non-abelian. The fourth size-21 PG(2, 4) magma magma#b904cba0 is the most rigid: |Aut| = 1 (trivial). Display reorder presents elements as a 3×7 grid: rows are the 3 orbits of an order-7 automorphism σ_7, and within each row elements are ordered along the σ_7-cycle starting from the minimum-canonical-label element. Under this reorder the Cayley table has a row-shift symmetry: T_new[shift(x), shift(y)] = shift(T_new[x, y]) where shift increments column index mod 7 within each 7-block — directly reflecting the σ_7 cyclic action. [text written by Claude]