Normalizers of subsystem subgroups in finite groups of Lie type

Abstract

ABSTRACT. In the present paper normalizers of subsystem subgroups of finite groups of Lie type are found in terms of groups of their induced automorphisms.

On the number of classes of conjugate Hall subgroups in finite simple groups

“…, and L 0 = L 0 ∩ L. The subgroup L 0 is a reductive subgroup of maximal rank of L. By [30,Theorem 2] it follows that Aut L (L ρ 1 0 ) does not contain diagonal automorphisms, hence by using [19,Proposition 4.1.6], we obtain the statement of the lemma for n odd. Lemma 2.12 Assume that a simple classical group G and its subgroup H of satisfy one of the following statements:…”

“…So, for every subgroup L 0 of L, stabilizing the decomposition V 1 ⊥ V 2 , there corresponds a unique subgroup L 0 of L, stabilizing the decomposition 2 ), and L 0 = L 0 ∩ L. The subgroup L 0 is a reductive subgroup of maximal rank of L. By [30,Theorem 2] it follows that Aut L (L ρ 1 0 ) does not contain diagonal automorphisms, hence by using [19,Proposition 4.1.6], we obtain the statement of the lemma for n odd. P Lemma 2.12.…”

“…(a) G PSL 2 (q), (q, 6) 30) [4] it also follows that Out(Ω 7 (2)) is trivial, while the universal central extension 2 . Ω 7 (2) of Ω 7 (2) has no faithful irreducible representations of degree 7 over F q .…”

Strong reality of finite simple groups

“…Then M ≃ SL 2 (q 3 ) • SL 2 (q), |Z(M)| = 2 and S = {e}. Moreover, |C : M| = 2 and, by [18,Theorem 2], C/ SL 2 (q) ≃ PGL 2 (q 3 ) and C/ SL 2 (q 3 ) ≃ PGL 2 (q). We write u as u 1 · u 2 , where u 1 ∈ SL 2 (q 3 ), u 2 ∈ SL 2 (q), and let v 1 , v 2 be the images of u 1 , u 2 in C/ SL 2 (q) and C/ SL 2 (q 3 ), respectively.…”