Equation 677 Database

Size-25 = 5² idempotent right-cancellative magma satisfying Eq 677 and Eq 255, in the AG(2, 5) line family. The 30 size-5 sub-magmas are exactly the lines of the affine plane AG(2, 5); every pair of distinct elements lies on a unique line, forming a Steiner system S(2, 5, 25). Distinguishing structural feature: |Aut(M)| = 20 (the smallest Aut among the 29 AG(2, 5) line magmas in DB). Aut acts on the 25 elements with orbit structure (1, 4, 20): • Singleton orbit {12}: a unique Aut-fixed element (the 'pivot'). • Size-4 orbit {5, 8, 16, 23}: 4 elements forming one Aut-orbit. • Size-20 orbit: the remaining 20 elements forming one Aut-orbit. Aut(M) contains an order-20 element σ with cycle structure (1, 4, 20) — fixes 12, acts as a 4-cycle on the size-4 orbit, and as a 20-cycle on the size-20 orbit. The structure |Aut| = 20 with this orbit pattern is reminiscent of AGL(1, F_5) (order 20) acting on F_5 ⊕ (something), but more rigid than the size-5 magma's natural Aut. Compare with magma#4aacf2e3 (|Aut| = 48, orbits (1, 24), cycle (1, 24)) and magma#09d21ec3 (|Aut| = 600 = AGL(1, F_25), sharply 2-transitive). This 4eb40a4b magma has even more rigid Aut than 4aacf2e3. Display reorder uses the Aut-orbit + σ-cycle structure: • Position 0: the pivot (12). • Positions 1-4: size-4 orbit in σ-cycle order (4-cycle of σ). • Positions 5-24: size-20 orbit as a 4×5 sub-grid where rows are σ⁴-orbits (4 cosets of an order-5 subgroup) and columns enumerate positions within each row via σ. Under this reorder the Cayley table shows a clean 'sub-block' structure: the upper-left 5×5 corner involves the pivot and size-4 orbit; the lower-right 20×20 block shows diagonal-banded structure from the σ-action on the 20-orbit. This is cleaner than the AG-grid (5×5 of 5×5 blocks) reorder for this specific magma because its small Aut group doesn't fully respect any AG(2, 5) parallel class as a 'symmetry axis'. [text written by Claude]