Why is M₁₂ the one happy-family group read both as a piece of the Golay object and as a centraliser factor in the Monster, and is that one fact or two?

Closed 2026-10-23, before a panel: the must-predict was settled by computation. The two M₁₂'s are not one subgroup. GAP's stored fusions (R28_reading/case33/RESULTS-gap.md, a lab chat on GAP 4.16.1 and CTblLib 1.3.11) and the codifier's lift check (R28_reading/case33/lift33.g, .log) show the following. - The centralising M₁₂ (inside (L₂(11) × M₁₂):2) has its two classes of order 3 in M-classes 3B and 3A, its 5-elements in 5A and its 11-elements in 11A. - The dodecad M₁₂ (M₁₂ → M₂₄ → 2¹¹:M₂₄ → Co₁) has its order-3 classes in Co₁-classes 3B and 3D. Each of those has exactly one class of order-3 elements above it in 2^{1+24}·Co₁, fusing to M-classes 3A and 3C. Its 5-elements reach 5A and its 11-elements 11A. - Any subgroup of 2^{1+24}·Co₁ ≤ M mapping isomorphically onto the dodecad M₁₂ therefore has elements of order 3 in class 3C, and the centralising M₁₂ has none. The outer automorphism swaps the two order-3 classes, so it cannot repair this: as sets, {3A, 3B} ≠ {3A, 3C}. The copies are not conjugate in M, whether or not the dodecad M₁₂ lifts at all. - So the double placement is two different subgroups with the same isomorphism type. The happy family's forest was drawn on isomorphism types, so its one overlap is an overlap of names. Whether the forest should be drawn on conjugacy classes of subgroups is PG-04's question, not a panel's.

Status: closed 2026-10-23 by computation (two non-conjugate M₁₂ subgroups: the centralising copy has its order-3 elements in 3A and 3B, the dodecad copy in 3A and 3C), before any panel. The draft header that stood here, written while the case was open, is in git.

---

type: derivation

field: math

object. The twenty sporadic groups that occur inside the Monster M (the "happy family"). Each group was read by an independent reader, who put it in one or more of four kinds: - W: the full symmetry group of a whole object, up to its centre; - V: the stabiliser of a piece of such an object; - R: the symmetry of a coordinatisation (a lattice over a larger ring); - F: the centraliser in M of an element or subgroup of M, up to its centre: C_M(x) = Z × G or Z.G with Z central.

friction. In two independent readings (runs 115 and 116, twenty fresh readers each), every group got the same set of kinds. Exactly one group got two kinds: M₁₂ was placed V (the setwise stabiliser of a dodecad of the Golay code, inside M₂₄) and F (C_M(11A) = 11 × M₁₂, up to centre). Every other group got exactly one kind. Is the double placement one fact seen twice, two unrelated facts, or an artefact of how F was counted?

background. - Measured (RESULTS116.md). The kinds table: W {M₂₄, Co₁, M}; V {M₁₁, M₁₂, M₂₂, M₂₃, Co₂, Co₃, HS, McL}; R {J₂, Suz}; F {M₁₂, Fi₂₂, Fi₂₃, Fi₂₄′, He, HN, Th, B}. M₁₂ is in both V and F. Every affirmed kind cites a URL. - Measured (RESULTS118.md). Among the prime-order classes of M whose centraliser has a sporadic factor up to centre (3C Th, 5A HN, 7A He, 11A M₁₂), 11A is the only one whose structure no reader found stated in a source. Two of three readers derived 11 × M₁₂ from the normaliser (11:5 × M₁₂):2 and the centraliser order 11·|M₁₂|. The third declined. - As counted. The readers' own reports name two centraliser placements that the F rule excluded, because there the group is a quotient of a non-central extension, not a factor modulo centre: Co₁ in C_M(2B) = 2^{1+24}·Co₁, and Suz in 3^{1+12}.2Suz.2 (RESULTS116.md, the Co₁ and Suz rows). - Checked by a reader, 2026-10-23 (R28_reading/case33/verify-reader.md; web only, given the claims and nothing else). - The maximal subgroup is (L₂(11) × M₁₂):2, not L₂(11) × M₁₂ as the run-116 row implied; L₂(11) × M₁₂ is its index-2 subgroup. N_M(11A) = (11:5 × M₁₂):2 < (L₂(11) × M₁₂):2 (Wilson 1988). - |C_M(11A)| = 11·|M₁₂|, and M has one class of order 11 (arXiv 2412.12182). "C_M(11A) ≅ 11 × M₁₂" is again derived from orders, not quoted: the third reader in a row to find it unstated. - Out(M₁₂) has order 2 and exchanges the two inequivalent 12-point actions. In M₂₄ the dodecad stabiliser is M₁₂, acting "differently on 2 sets of 12, reflecting the outer automorphism" (the complement's 5-transitivity is inferred, not quoted). - Co₁ has the maximal subgroup 2¹¹:M₂₄, and C_M(2B) = 2^{1+24}·Co₁. - N_M(3B) = 3^{1+12}.2Suz.2, with C_M(3B) = 3^{1+12}.2Suz. - Prime-order centralisers with a sporadic direct or central factor: 2A (2.B), 3A (3.Fi₂₄′), 3C (3 × Th), 5A (5 × HN), 7A (7 × He), 11A (11 × M₁₂). None at 2B, 3B, 5B, 13A, 13B, 29, 41, 59 or 71. Unclear for 7B and for 17, 19, 23, 31 and 47 (no centraliser structure found).

Coincidences this case leans on. (a) M₁₂ alone has two kinds. Partly explained (the narrower fact below checked on every prime-order class except 7B, 17, 19, 23, 31, 47), and not as the case first assumed. It holds under the F rule as sealed (up to centre only). Under the rule "G is a composition factor of a centraliser of a prime-order element of M", Co₁ (from 2B) would be W and F, and Suz (from the 3^{1+12}.2Suz.2 its reader cites) would be R and F, on the readers' own evidence. M₁₂ would be one of three. So the fact is narrower than "the single two-parent vertex": M₁₂ is the only group that is both a stabiliser inside the Golay/Leech objects and a direct factor, up to centre, of a prime-order centraliser in M. (b) The prime is 11, and 11 divides |M₁₂| (an 11-cycle on twelve points fixes one point): unexplained. Whether the 11-element C_M(11A) is built from relates to the 11-cycles M₁₂ contains has not been checked. (c) Both runs agree on every group: explained as replication of the kinds. It is not independent evidence about M₁₂.

Must predict. The decision this would shape is PG-04's next seal, the pre-registration's "why M₁₂" guesses 2 and 3, which are not yet designed. It is not a pending run. A satisfactory answer predicts, as a computation GAP or the ATLAS can settle, whether the M₁₂ of 11 × M₁₂ ≤ M and the dodecad stabiliser M₁₂ ≤ M₂₄ are carried into each other by the route M ⊃ C_M(2B) = 2^{1+24}·Co₁ → Co₁ ⊃ 2¹¹:M₂₄ → M₂₄ (Co₁ ⊃ 2¹¹:M₂₄ and C_M(2B) = 2^{1+24}·Co₁ checked by the reader). In other words: are the two placements conjugate images of one subgroup along that route, or are they unrelated copies? A take that predicts "unrelated" must also say what then makes the double placement non-accidental, or say that it is accidental.

Not part of the case

• The codifier's error, found by writing the case. "M₁₂ the single two-parent vertex" was carried into the programme and the pre-registration as if it were a fact about M₁₂. It is a fact about M₁₂ under a counting rule: the F rule draws its line between central and non-central extensions, and that line is ours, not the groups'. The run-115/116 seal chose it, and nobody asked whether the forest's one overlap survives moving it. Before the panel, restate (a) as the narrower fact and check that it holds: that no other group is both V or R and a direct factor up to centre. The table supports it, but it was never asked that way.
• Leads stay in docs/walk/2026-10-21-pre-registration-symmetry.md (section of 09:45). They are not repeated here. Panels never see the pre-registration.