Finite linear groups of small degree

Summary

Abstract

This paper is a survey of results in the study of finite subgroups of low-dimensional classical groups. The author discusses mainly his classification of the conjugacy classes and the normalizers of finite absolutely irreducible quasisimple linear groups over finite and algebraically closed fields up to degree 27.

Our aim is to discuss recent developments in the following classical problem.

Problem 1Describe the finite linear groups of small degree, i. e. finite subgroups in GL_n(K) for every field K and small n.

Beginning in the middle of the last century, this problem attracted the attention of many mathematicians. By the seventies of our century it was solved for K = ℂ and n ≤ 9 in the papers by Jordan, Klein, Valentiner, Blichfeld (see [4]), Brauer [7], Lindsey [41], Huffman and Wales [16], [17], and Feit [13]. The case char K = p > 0 and n ≤ 5 of Problem 1 was considered before 1982 in the papers by Moore [44], Burnside [8], Wiman [52], Dickson [10], [11], Mitchell [43], Hartley [15], Bloom [5], Mwene [45], [46], Wagner [50], Di Martino and Wagner [12], Zalesskii [53], [54], Suprunenko [49], and Zalesskii and Suprunenko [57]. For finite K this problem stands as Problem 40 in the list of important problems of group representation theory formulated by Brauer in his lectures [6]. Results related to Problem 1 (see, for example, the surveys of Zalesskii [55], [56] and the author [26]) have found numerous applications, in particular, in the classification of finite simple groups (CFSG).