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)| = **1** (trivial). This is the MOST RIGID size-21 PG(2, 4) magma in the DB — no non-trivial symmetries at all. Compare with the other three size-21 PG(2, 4) magmas: • magma#b1cfacfa: |Aut| = Z_21 cyclic (regular) • magma#4bd29022: |Aut| = F_21 = Z_7 ⋊ Z_3 (non-abelian Frobenius) • magma#50e6ad54: |Aut| = F_21 (another non-abelian Frobenius) • magma#b904cba0 (this one): |Aut| = 1 All four magmas realize the same PG(2, 4) line structure but with progressively less algebraic symmetry — this one is at the extreme end, with no element distinguished from any other by any automorphism. Yet the magma operation isn't arbitrary: it must respect the PG(2, 4) Steiner system AND satisfy Eq 677. Structural uniformity despite trivial Aut: for EVERY element x, the cycle structure of L_x is (1, 2¹⁰) (= fixed-point-free involution on M \ {x}, plus x fixed) and the cycle structure of R_x is (1, 4⁵) (= order-4 permutation with 5 cycles of 4 + x fixed). So while no automorphism swaps elements, every element 'looks the same' combinatorially. Display reorder uses an AG(2, 4) + line-at-infinity layout (= the standard 'embedding' of AG(2, 4) into PG(2, 4) by removing one line): • Positions 0-4: a chosen line L_∞ (5 points). Specifically, the line through element 0 with min 2nd-smallest member. • Positions 5-20: the remaining 16 affine points of AG(2, 4), laid out as a 4×4 grid using two chosen parallel classes (directions) of AG(2, 4) as the row and column coordinates. Each PG(2, 4) line not equal to L_∞ becomes an AG(2, 4) line of 4 points (sharing 1 'direction' point on L_∞). Under this reorder the rendered Cayley table has a 5+16 partition visible: the top-left 5×5 corner involves the 'line at infinity', the lower-right 16×16 sub-block shows the affine-plane operation with 4×4 sub-structure from the parallel-class directions, and the L-shaped corner pieces (5×16, 16×5) reflect cross-region operations. [text written by Claude]