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]
dwrensha · 2026-05-15 11:16:54
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 of this magma are exactly the lines of the affine plane AG(2, 5); every pair of distinct points lies on a unique line (so this realizes the Steiner system S(2, 5, 25)). Each line is isomorphic to magma#e549b5f8 (the unique size-5 Eq 677 magma over F_5, x ◇ y = 2x + 4y mod 5 — α = 4 is the only primitive 10th root of unity mod 5).
The 30 lines decompose into 6 parallel classes of 5 mutually disjoint lines each (covering the 25 elements). Each parallel class is one of the q + 1 = 6 'directions' of AG(2, 5).
Display reorder uses an AG(2, 5) coordinate grid: pick two parallel classes C_h and C_v, sort each class's lines by minimum canonical label, then label each point as (i, j) at index 5·i + j where i is its line-index in C_h and j its line-index in C_v. Each pair of lines from different classes intersects in exactly one point, so this gives a bijection between elements and (Z/5) × (Z/5) coordinate positions. Under this reorder the Cayley table shows a clean 5×5 grid of 5×5 blocks: the 5 diagonal blocks are within-class C_h line operations (each block is the size-5 magma's Cayley table), and the 20 off-diagonal blocks reflect the cross-line operations.
There are 29 size-25 magmas in this AG(2, 5) line family currently in the DB. Of these, only 1 (magma#053cceeb, the F_5 × F_5 direct product with both factors using α=4) is fully (Z/5)²-translation-invariant; the other 28 share the same AG(2, 5) Steiner-system structure but their magma operations break the additive translation symmetry — they have rigid automorphism groups (typically order 24-120) with no order-5 fix-free element, so they are not 'linear' over F_5² in the additive sense.
Other size-25 Eq 677 magma families in the DB (NOT in this family): (a) 23 'F_5(2, 4) base × F_5 fiber' fiber-bundle constructions with only 5 size-5 sub-magmas (different from the 30 here); (b) magma#d732efd1, a 'half-AG(2, 5)' sporadic non-product magma with 15 size-5 sub-magmas; (c) magma#09d21ec3, a sporadic non-product magma with NO size-5 sub-magmas at all.
[text written by Claude]
dwrensha · 2026-05-15 11:32:59
dwrensha · 2026-05-15 11:16:54