Local subgroups of the Monster and odd code loops. (English) Zbl 0848.20010
Using loops built from a self-orthogonal code over a field of order \(p\), \(p=3,5,7\), the author constructs normalizers in the sporadic simple group \(F_1\) of certain elementary abelian subgroups of orders 9, 25, and 49. These normalizers are of shapes \(3^{2+5+10}\cdot(M_{11}\times\text{GL}(2,3))\), \(5^{2+2+4}\cdot(S_3\times\text{GL}(2,5))\), and \(7^{2+1+2}\cdot\text{GL}(2,7)\). A similar method was used earlier by J. Conway [Invent. Math. 79, 513-540 (1985; Zbl 0564.20010)] to construct some 2-local subgroup of \(F_1\).
Reviewer: V.Mazurov (Novosibirsk)
Keywords:
\(p\)-local subgroups; Monster; loops; self-orthogonal codes; normalizers; sporadic simple group \(F_ 1\); elementary Abelian subgroupsCitations:
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