On Frobenius groups. II: Universal completion of nearfields of finite degree over a field of reference
In the 1930's the author characterizing the complements in Frobenius groups [see Abh. Math. Semin. Hamb. Univ. 11, 187-220 (1935; Zbl 0011.10302)] put a milestone in the basement of the modern finite group theory. The suitable language in this paper included near-fields: fields were characterized as commutative complete near-fields, and all finite non-Dickson complete near-fields, 7 in number, were found. (A near-field is said to be complete if the left multiplication is defined for each element as a left multiplier.) In this paper the author constructs a new class of complete non-Dickson near-fields which are universal in some sense. The starting point is some subgroup \(G_0\) of \(\mathrm{GL}(n,k)\) which is fixed-point-free in the sense that only the identity element of \(G_0\) has eigenvalue 1. There is also an application to the problem of Biliotti about Dembowski's generalization of Hughes planes. [For part I see ibid. 5, 132--145 (1985; Zbl 0591.20017).]
scientific article; zbMATH DE number 4059543