On a theorem of Goldschmidt. (English) Zbl 0632.05032

Weiss, Richard

Ann. Math. (2) 126, 429-438 (1987).

Both P. Delgado, D. M. Goldschmidt and B. Stellmacher [Groups and graphs: New results and methods (1985; Zbl 0566.20013)] and B. Stellmacher [Arch. Math. 46, 8-17 (1986; Zbl 0588.20013)] used a so- called “pushing up” approach (see latter reference for definition) in shortening D. M. Goldschmidt’s original proof [Ann. Math. 111, 377- 407 (1980; Zbl 0475.05043)] of his theorem: Let \(\Gamma\) be a connected trivalent graph. Let \(\{\) x,y\(\}\) be an edge and G a group of automorphisms of \(\Gamma\) acting edge-transively such that \(G_ x\) is finite. Then \(| G_{xy}|\) divides \(2^ 7\). The present paper offers a simple proof that does not use pushing-up.

Reviewer: M.E.Watkins

MSC:

05C25	Graphs and abstract algebra (groups, rings, fields, etc.)
20B25	Finite automorphism groups of algebraic, geometric, or combinatorial structures
20B27	Infinite automorphism groups

Keywords:

stabilizer; pushing up; trivalent graph; edge-transively

Citations:

Zbl 0566.20013; Zbl 0588.20013; Zbl 0475.05043