The maximal subgroups of the Chevalley groups F4(F) where F is a finite or algebraically closed field of characteristic not equal to 2,3

Citation

Magaard, Kay (1990) The maximal subgroups of the Chevalley groups F4(F) where F is a finite or algebraically closed field of characteristic not equal to 2,3. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/D2GB-VK65. https://resolver.caltech.edu/CaltechETD:etd-06132007-094324

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. We find the conjugacy classes of maximal subgroups of the almost simple groups of type F4(F), where F is a finite or algebraically closed field of characteristic not equal to 2,3. To do this we study F4(F) via its representation as the automorphism group of the 27-dimensional exceptional central simple Jordan Algebra J defined over F. A Jordan Algebra over a field of characteristic not equal to 2 is a nonassociative algebra over a field F satisfying xy = yx and [...] = [...] for all its elements x and y. We can represent Aut(F4(F)) on J as the group of semilinear invertible maps preserving the multiplication. Let G = F4(F) and [...]. We have defined a certain subset of proper nontrivial subalgebras as good. The principal results are as follows: SUBALGEBRA THEOREM: Let F be a finite or algebraically closed field of characteristic not equal to 2,3. Let H be a subgroup of [...] and suppose that H stabilizes a subalgebra. Then H stabilizes a good subalgebra. The conjugacy classes and normalizers of good subalgebras are also given. STRUCTURE THEOREM: Let H be a subgroup of [...] such that [...] is closed but not almost simple. Then H stabilizes a proper nontrivial subalgebra or H is contained in a conjugate of [...]. The action of [...] on J is described and it is shown that [...] is unique up to conjugacy in G. THEOREM : If L is a closed simple nonabelian subgroup of G, then [...] is maximal in [...] only if L is one of the following: [...]. For each member [...] we identify those representations [...] which could give rise to a maximal subgroup of G and show the existence of [...] in G. Up to few exceptions we also determine the number of G conjugacy classes for each equivalence class [...].

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