Maximal subloops of finite simple Moufang loops.
type: Item
instance of: scholarly article
title: Maximal subloops of finite simple Moufang loops.
zbMATH Open document ID: 1120.20065
author: Alexandre N. Grishkov
author: A. V. Zavarnitsine
published in: Journal of Algebra
publication date: 2006-12-07T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 17A75
Mathematics Subject Classification ID: 20D06
zbMATH DE Number: 5078141
zbMATH Keywords: finite simple Moufang loops
zbMATH Keywords: non-associative Moufang loops
zbMATH Keywords: groups with triality
zbMATH Keywords: Cayley algebras
zbMATH Keywords: maximal subloops
zbMATH Keywords: simple alternative algebras
zbMATH Keywords: automorphism groups
reviewed by: Huberta Lausch
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.02.038
OpenAlex ID: W1963724920
cites work: Lagrange's theorem for Moufang loops
cites work: GROUPS WITH TRIALITY
cites work: Simple Moufang loops
cites work: Q3947798
cites work: The classification of finite simple Moufang loops
cites work: The maximal subgroups of the finite 8-dimensional orthogonal groups \(P\Omega ^ +_ 8(q)\) and of their automorphism groups
cites work: Q5515384
cites work: Homotopes of alternative algebras
cites work: Q3290212
cites work: Q3996618
cites work: Q5333669
cites work: Some Geometry of Triality with Applications to Involutions of Certain Orthogonal Groups
cites work: Moufang Loops of Small Order. I
cites work: The maximal subloops of the simple Moufang loop of order 1080.
cites work: Q5732712
cites work: The characterization of \(\text{PSL}(2,q)\) by its element orders
cites work: Integral Cayley numbers
cites work: Q3996420
cites work: Q2774024
cites work: Automorphism groups of simple Moufang loops over perfect fields
cites work: The maximal subgroups of the Chevalley groups \(G_ 2(q)\) with q odd, the Ree groups \(2G_ 2(q)\), and their automorphism groups
cites work: Maximal subgroups of \(G_2(2^n)\)
cites work: Q4488076
cites work: Q4930353
DOI: 10.1016/J.JALGEBRA.2006.02.038
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description: scientific article; zbMATH DE number 5078141