2019/04/24 by Tom De Medts, De Medts, Tom, Karsten Naert +1
Mathematics · #20D06 #20G15 #20G40 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1904.10741
openalex publication_date 2019/04/24 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
Among the infinite classes of finite simple groups, the most exotic classes\nare probably the Suzuki groups and the Ree groups. They are "twisted versions"\nof groups of Lie type, but they cannot be directly obtained as groups of\nrational points of a suitable linear algebraic group. We provide a framework in\nwhich these groups do arise as groups of rational points of algebraic groups\nover a "twisted field"; in the finite case, such a twisted field can be\ninterpreted as a "field with \√( smash[b]p) elements".\n Our framework at once allows for other, perhaps less known, exotic families\nof groups. Most notably, there is a class of "mixed groups", introduced by J.\nTits but also apparent in the work of Steinberg, and we show that they can be\nobtained as groups of rational points of algebraic groups over a "mixed field".\n We show that a base change from mathbbF\√( smash[b]p) to\n mathbbFp transforms twisted groups into mixed groups, and we formulate a\nnotion of "twisted descent" that allows to detect which mixed groups arise in\nthis fashion.\n