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]
dwrensha · 2026-05-15 11:16:59
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 11:37:06
dwrensha · 2026-05-15 11:16:59