Pure quantitative characterization of linear groups over the binary field

Abstract

The purpose of this paper is to discuss the pure scalar characterization of the automorphism group Aut(L 5 (2)) and the linear group L 6 (2). It is proved that Aut(L 5 (2)) and L 6 (2) can be characterized quantitatively by the set of element orders. The main results are obtained by using William's work on prime graph components of finite groups and Brauer characters in trivializing the possible 2-subgroups.

Recognition by Spectrum of Some Linear Groups over the Binary Field

“…(15) If P ∼ = 2 D n (3), 9 ≤ n = 2 m + 1 = r, where r is an odd prime number, then 3 n−1 +1 2 = 2 p − 1 and we get a contradiction similar to (6). (16) Suppose that P ∼ = G 2 (q), 2 < q ≡ ε mod 3, ε = ±1.…”

“…Table 1 G s(G) h(G) References L 3 (2) ∼ = L 2 (7) 3 1 [3] L 4 (2) ∼ = A 8 2 1 [4] L 5 (2) 2 1 [5] L 6 (2) 2 1 [5,6] L 7 (2) 2 1 [5,7] L 8 (2) 2 1 [8] L 9 (2) 1 Unknown…”

“…This is just because our method of recognition in concerned with the groups whose Gruenberg-Kegel graph is disconnected and the two groups L 9 (2) and L 10 (2) have connected Gruenberg-Kegel graphs. So, we can pose the following open problem (also see [6]). …”