Thompson–wielandt-Like Theorems Revisited

Abstract

This paper provides a unified and elementary proof of four Thompson–Wielandt‐like theorems.

On the stabilisers of locally 2-transitive graphs

“…We notice that PSL 2 (7) ∼ = SL 3 (2) ∼ = GL 3 (2). Then soc(G vw ) = PSL 2 (7), and G v , G w ✄ A 7 or 2 3 :GL 3 (2). Therefore, we conclude that {G v , G w } = {A 7 , 2 3 :GL 3 (2)}, which is listed in row 14 of Table G vw soc(G v ) q n :SL n (q).O PSL n+1 (q) SL n (q).O q n Sp 2n (q).O q 2n 2 1+4 :A 5 , 2 1+4 :S 5 The candidate in the first line or the third line is not isomorphic to any other one in the table.…”

“…Assume that soc(G vw ) is a linear group which is not isomorphic to A m with m 5. We notice that PSL 2 (7) ∼ = SL 3 (2) ∼ = GL 3 (2). Then soc(G vw ) = PSL 2 (7), and G v , G w ✄ A 7 or 2 3 :GL 3 (2).…”

On G-locally primitive graphs of locally Twisted Wreath type and a conjecture of Weiss

“…The Thompson-Wielandt theorem [8,22] is an essential tool in the proof of many results concerning with the Weiss Conjecture and the proof of Theorem 1 relies on it. In the version in [2,16], this important theorem states that, if Γ is G-locally primitive and (u, v) is an arc of Γ , then G [1] uv is a p-group for some prime p. Before we prove Theorem 1 we establish the following lemma, which is inspired by the structure theorem for primitive groups of TW type in [1, Section 5].…”

Two local conditions on the vertex stabiliser of arc-transitive graphs and their effect on the Sylow subgroups

“…A first generalisation of the theorem of Thompson was obtained by Wielandt [17,Theorem 6.6]. In graph theoretic terminology, the Thompson-Wielandt theorem states that for a Gvertex-primitive digraph ∆, if (x, y) is an arc of ∆, then the subgroup of G fixing pointwise ∆(x) and ∆(y) is a p-group for some prime p. This result inspired much research and various generalisations of the so called Thompson-Wielandt theorems have been obtained by many authors [4,6,14,16].…”

“…Section 2 consists of seven technical lemmas whose proof is obtained by adapting, to the more general context of Hypothesis 1, Lemmas 2.1-2.7 in [14]. Since the paper is short and elementary, we give a full argument although the proofs of our lemmas are essentially as in [14]. The proofs of Theorem 2 and Corollary 3 are in Section 3.…”