Recognizability of finite simple groups L 4(2m) and U 4(2m) by spectrum

Abstract

It is proved that finite simple groups L 4 (2 m ), m 2, and U 4 (2 m ), m 2, are, up to isomorphism, recognized by spectra, i.e., sets of their element orders, in the class of finite groups. As a consequence the question on recognizability by spectrum is settled for all finite simple groups without elements of order 8.

Recognition by spectrum for finite simple groups of Lie type

“…Groups L 16 (2 m ) were proved to be recognizable by the same authors and Shi [10]. Quite recently, Chen and Mazurov [15] showed that groups L 4 (2 m ) and U 4 (2 m ) are recognizable for m > 1.…”

Recognition by spectrum for finite linear groups over fields of characteristic 2

“…For n 4, as shown in [2][3][4][5], the group L n (2 k ) is recognizable, and hence the theorem is valid in this instance. Therefore, we may assume that n 5.…”

“…For groups L = L n (2 k ), the recognizability problem is solved for n = 2 (see [2]), n = 3 (see [3,4]), n = 4 (see [5]), n = 16 (see [16]), n = 2 m 32 (see [7]), and for k = 1 (see [8,9]); in all these cases L turns out to be recognizable by spectrum. As follows from [10,11], for n > 5, the group L = L n (2 k ) is recognizable by spectrum among its covers, i.e., among finite groups containing L as a homomorphic image (see Lemma 5).…”

Recognition by spectrum for finite simple linear groups of small dimensions over fields of characteristic 2

“…For simple linear groups L n (2 k ), the recognizability problem is solved with n = 2 [3], n = 3 [4,5], n = 4 [6], 11 n 17 [7,8], n 26 [8,9], and also for k = 1 [10,11]. The goal of the present paper is to solve the problem for all the remaining groups L n (2 k ), thus settling the question of whether finite simple linear groups over fields of characteristic 2 are recognizable by spectrum.…”