Equation 677 Database

Size-25 idempotent right-cancellative magma satisfying Eq 677 and Eq 255, in the AG(2, 5) line family. Every pair of distinct elements generates a 5-element sub-magma (≅ unique size-5 Eq 677 magma magma#e549b5f8); the 30 such sub-magmas are exactly the 30 lines of the affine plane AG(2, 5), partitioned into 6 parallel classes of 5 disjoint lines each — a Steiner system S(2, 5, 25). Distinguishing structural feature: |Aut(M)| = 48, acting on the 25 elements with orbit structure (1, 24) — there is a UNIQUE Aut-fixed element (canonical label 16 in the original labeling) and the other 24 elements form a single Aut-orbit. So this magma has a 'distinguished pivot' analogous to the size-76 pencil magma#875876e7, but at much smaller scale: 25 = 1 + 24 = 1 + 6·4 — one pivot + the 6 lines of AG(2, 5) through the pivot, each contributing 4 non-pivot elements. The Aut group contains an order-24 element σ that fixes the pivot and acts as a single 24-cycle on the other 24 elements. Combined with σ²=24/2=12, etc., the structure (Z_24 acting transitively on 24-orbit, with the multiplicative stabilizer of an element being Z_2) is reminiscent of AGL(1, F_25)/Z_2 — exactly half the sharply 2-transitive group of order 600 that acts on the size-25 exceptional Zassenhaus near-field magma#09d21ec3. Compare also magma#d732efd1 (half-AG(2,5) non-RC) which has |Aut| = 300 (= 25 · 12), also half of the full near-field's 600. Display reorder: place the pivot at position 0, then enumerate the other 24 elements as σ⁰(x), σ¹(x), σ²(x), …, σ²³(x) for some starting x and the order-24 cyclic automorphism σ. Under this reorder the Cayley table shows clean diagonal banding from the cyclic σ-action: every row (except row 0) is a shift of the previous row by 1 in the cyclic index. NOT (Z/5)²-translation-invariant in any abelian-group sense: this magma has no fix-free order-5 magma automorphism, so its 25-element carrier doesn't admit a regular abelian translation subgroup. The 'symmetry' is concentrated in the multiplicative cyclic Z_24 around the fixed pivot rather than additive translations. [text written by Claude]