Equation 677 Database

Size-25 = 5² idempotent right-cancellative magma satisfying Eq 677 and Eq 255. **The exceptional Zassenhaus near-field of order 5²** — the smallest member of the exceptional-near-field family. Structure: • Additive carrier: (Z/5)² ≅ GF(25) (the additive group of F_25). The magma is fully (Z/5)²-translation-invariant — verified by exhibiting two commuting order-5 fix-free magma automorphisms τ_1, τ_2. • Automorphism group: Aut(M) is the sharply 2-transitive group AGL(1, F_25) = (Z/5)² ⋊ SL(2, 3) of order 25 × 24 = 600 (= |F_25*|). For Zassenhaus's exceptional near-field at order p² = 25, the multiplicative group of order 24 is SL(2, 3) (the binary tetrahedral group). Aut acts transitively on the 25 elements. • Operation: x ◇ y = x + f(y − x) on (Z/5)², where f is multiplication by a near-field element c of order 8 in the SL(2, 3) multiplicative group (L_0 has cycle structure (1, 8³): order 8, with 3 orbits of length 8 partitioning the 24 non-zero elements into 3 cosets of ⟨c⟩). Distinguishing features: • EVERY pair of distinct elements generates the full 25-element magma (no proper sub-magmas of any size). This is uniquely different from the other 53 size-25 magmas: all the others have size-5 sub-magmas (AG(2, 5) family) or size-5 fibers (the F_5(2,4)×F_5 family) or partial-AG structure. This one has NO size-5 sub-magmas because the non-distributive near-field multiplication breaks the F_5-linearity required for size-5 lines to exist as sub-magmas. • L_0 cycle structure (1, 8³): order 8, 3 cosets of an order-8 cyclic subgroup of the (non-commutative) multiplicative group SL(2, 3). • In the (Z/5)² basis (chosen via τ_1, τ_2 commuting order-5 automorphisms): f is axis-additive (linear on each F_5-axis) but NOT globally F_5-linear — verified e.g. f(1,0) + f(0,1) = (4,3) + (2,2) = (1,0) ≠ (0,1) = f(1,1). This 'axis-additive but globally non-linear' fingerprint is the structural signature of near-field multiplication outside the prime subfield, identical to magma#5ebfbb80 (size 11², exceptional near-field of order 121) and magma#29114da6 (size 29², exceptional near-field of order 841). Display reorder presents elements as (i, j) ↦ τ_1^i τ_2^j(0) at index 5·i + j, where τ_1, τ_2 are the chosen commuting order-5 magma automorphisms (the hidden additive translations by '1' and '2' in (Z/5)²). Under this reorder the table is fully (Z/5)²-translation-invariant. [text written by Claude]