Some aspects of finite linear groups: A survey
Finite linear groups have been studied since the beginning of the 20th century. Of permanent attention were the classifications of (A) groups of small degrees and (B) groups generated by a conjugacy class of particular elements. The results, obtained without the classification of finite simple groups (CFSG), were partially reflected in the surveys of \textit{W. Feit} [Actes Congr. int. Math. 1970, Vol. I, 55-93 (1971; Zbl 0344.20008)], \textit{H. S. Leonard} [``Finite simple groups'', Oxford 1969, 191-197 (1971; Zbl 0221.20014)], and \textit{A. E. Zalesskii} [Usp. Mat. Nauk 36, No. 5(221), 57-107 (1981; Zbl 0475.20029)]. The goal of the present survey is to discuss recent results, obtained mostly in the 90's and using the CFSG. In Sections 2, 6, 7 and 8, the authors report on new results in the line of (A). These results rely upon CFSG and upon new developments in the representation theory of quasi-simple groups. Among these are refinements of the Landázuri-Seitz lower bounds (cf. Sections 3, 4), and a classification of low-dimensional representations of finite groups of Lie type in cross characteristic (cf. Section 5). Sections 9 and 10 present new results in the line of (B): classifications of linear groups containing elements with specific minimal polynomial, respectively containing elements with simple spectrum. Finally, Sections 11, 12 and 13 give insight into some closely related activities. In particular, the authors consider the following questions for quasi-simple groups: when can the tensor product of two representations be irreducible, and which complex representations can stay irreducible modulo every prime?
representations of finite quasi-simple groups
scientific article; zbMATH DE number 1506086
scientific article