Equation 677 Database

Size-49 "pencil" magma — 8 size-7 sub-magmas all sharing a unique idempotent, in 2 iso classes (7+1). Size 49 = 7^2. Exactly 1 idempotent (element 48 in canonical labels), 48 non-idempotents. |Aut(M)| = 252 = 4 * 63 = 6 * 42 acting in 3 orbits: - {48} (the lone idempotent, fixed by all Aut) - 1 orbit of size 6 (the non-pivot part of the "special" line) - 1 orbit of size 42 (the non-pivot parts of the other 7 lines, joined into a single transitive action) There are 8 distinct size-7 sub-magmas in M ("lines through the origin"), each containing element 48 plus 6 other elements. Together with 48 they partition the 49 elements as 1 pivot + 8 lines * 6 non-pivots = 1 + 48 = 49. So this is an "8-line pencil through the unique idempotent" structure analogous to AG(2, 7)-lines-through-origin (which is the geometric source, although the magma is NOT a faithful AG(2,7)-line construction - see below). The 8 lines split into 2 iso classes (NOT all the same, contrary to a literal AG(2,7) interpretation): - 7 lines (the 42-orbit ones): each iso to magma#7981e2df, the size-7 Eq 677 magma with exactly 1 idempotent. Lines (in canonical labels): [0, 6, 12, 19, 24, 32, 48], [1, 7, 13, 18, 25, 31, 48], [2, 9, 15, 21, 27, 33, 48], [3, 10, 16, 22, 28, 34, 48], [4, 11, 17, 23, 29, 35, 48], [5, 8, 14, 20, 26, 30, 48], [36, 37, 38, 39, 40, 41, 48] - 1 "special" line (the 6-orbit one): iso to magma#baf8b55c, the *other* size-7 Eq 677 iso class: [42, 43, 44, 45, 46, 47, 48] Both iso classes have exactly 1 idempotent (in each sub-magma, the global idempotent 48 is the unique idempotent), so the lines all "share the unique idempotent". But the two size-7 iso classes are non-isomorphic as magmas, so the pencil is INHOMOGENEOUS - 7 lines of one type + 1 line of another. In the suggested reorder, the 49 elements are arranged as 8 consecutive 6-element blocks (positions 0..5, 6..11, ..., 42..47) one per line, with the unique idempotent 48 at position 48 (the very last). The diagonal 6x6 blocks reveal each line's operation on its 6 non-pivot elements. The "AG(2, 7)-pencil" description (in the original comment) captures the geometric intuition: AG(2,7) has 49 points and 8 lines through the origin (one per direction). Here, the magma has the same combinatorial incidence pattern - 8 lines through a single point - but with an asymmetry that makes one line iso-distinct from the other 7, consistent with |Aut| = 252 < |AGL(2, 7)| = 2,016 (the full affine group of AG(2,7)). So the magma is a *twisted* AG(2,7) pencil, not the canonical one. The original comment said "8 size-7 F_7(4,3) sub-magmas all... corresponding to the 8 lines through the origin" - the line-through-origin geometric picture is correct, but the claim that all 8 lines are iso to F_7(4,3) is incorrect: 7 are iso to magma#7981e2df, 1 to magma#baf8b55c (the special line {42..47, 48}). [text written by Claude]