Maximal subloops of finite simple Moufang loops.
Moufang loops are closely related to groups with triality. The authors [in Math. Proc. Camb. Philos. Soc. 139, No. 1, 41-57 (2005; Zbl 1091.20039)] revealed a correspondence between the subloops of the unique finite simple non-associative Moufang loop \(M(q)\) and certain subgroups of the simple group with triality \(P\Omega^+_8(q)\). Now they bring the simple alternative algebra \(\mathbb{O}(q)\) and its automorphism group \(G_2(q) \) into consideration. The geometry of triality related to the algebra is used to define the triality automorphisms of \(P\Omega^+_8(q)\) explicitly. The purpose of this paper is the classification of all maximal subloops of finite simple non-associative Moufang loops up to conjugacy with respect to automorphisms. As a corollary a description of the maximal subloops of the simple Moufang loop \(M(q)\), \(q=p^n\), is obtained. The maximal subloops are: \(\text{PSL}_2(q)\) for \(q^2\), \(q\) arbitrary; \(M(\text{PSL}_2(q),2)\) for \(q\neq 3\); \(M(q_0)\), \(q=q^k_0\) with \(k\) prime and \((q,k)\neq(\text{odd},2)\); \(\text{PGL}(\mathbb{O}(q_0))\) with \(q=q^2_0\) odd; \(M(2)\) with \(p=q\) odd. Moreover, all isomorphic maximal subloops of \(M(q)\) are conjugate by the group \(\text{Inn}(M(q))\) of inner automorphisms. The authors also determine the structure of the normalizer in \(\text{Inn}(M(q))\cong G_2(q)\) of each maximal subloop of \(M(q)\) and the explicit number of subloops of each type.
scientific article; zbMATH DE number 5078141
scientific article