Variations on a theme of Glauberman
Wedderburn published his pioneering result, known as Wedderburn's little theorem [\textit{J. H. Maclagan-Wedderburn}, Trans. Am. Math. Soc. 6, 349--352 (1905; JFM 36.0139.01)], proving that ``Any finite associative division ring is a field''. Ever since, this result have attracted many research interest within the area of non-associative rings. Indeed, during the past nine decades some contributions on that subject have improved our inderstanding of the category of all finite division rings, namely the Artin-Zorn theorem [\textit{M. Zorn}, Abh. Math. Semin. Univ. Hamb. 8, 123--147 (1930; JFM 56.0140.01)] which affirms that ``Any finite alternative division ring is a field'' and also its extension by Albert to the power-associative rings of characteristic $\neq 2, 3, 5$ [\textit{R. D. Schafer}, An introduction to nonassociative algebras. New York and London: Academic Press (1966; Zbl 0145.25601), Theorem 5.1]. The Artin-Zorn theorem also has an equivalent geometric counterpart claiming that ``Any finite projective plane satisfying little Desargues' theorem is Pappian'' [\textit{F. W. Levi}, Finite geometrical systems. Calcutta: University of Calcutta (1942; Zbl 0060.32304)]. \textit{G. Glauberman} [Proc. Lond. Math. Soc. (3) 25, 253--287 (1972; Zbl 0242.20018)] showed his characterization of finite fields of odd order, claiming that ``A subset $S$ of invertible matrices over finite field of odd order containning the identity matrix forms, together with the zero matrix, a field under matrix addition and matrix multiplication if and only if $S$ is closed under inversion and $S\cup\{0\}$ is closed under addition and scalar multiplication''. \textit{G. Heimbeck} [Elem. Math. 39, 95--98 (1984; Zbl 0551.12016)] extended the Glauberman's characterization to finite fields of even order. He showed the so-called Glauberman-Heimbeck Theorem claiming the following: ``Let $V\neq\{0\}$ be a finite vector space and $M$ a subset of $\mathrm{GL}(V)$ which is closed under inverses, has $M\cup\{0\}$ closed under addition and contains the identity. Then $M\cup\{0\}$ is a field''. In the paper under review, the authors exploit the connection between group theory, projective geometry and the theory of alternative division rings, to give a new proof of the \textit{Artin-Zorn Theorem}. They use elementary techniques of linear algebra (Theorems 4.1, 4.2, 4.3) and apply \textit{Glauberman-Heimbeck Theorem} (Theorem 3.2). Also, they extend, without use of geometry, the above \textit{Glauberman-Heimbeck characterization} for finite field to a larger class of fields. This required a laborious study (Theorems 5.1, 5.2, 5.4, 5.5, 5.7). All known examples of sets of matrices of maximal dimension satisfying \textit{Glauberman's conditions} over a general field that are not associative division rings are characterised (Theorem 5.7) via the \textit{Bruck-Kleinfeld-Skornyakov classification of alternative division rings} (Theorem 2.7).
scientific article; zbMATH DE number 7220545
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