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OnCSQ-normal subgroups of finite groups

Yong Xu;  Xianhua Li — 2016

Open Mathematics

We introduce a new subgroup embedding property of finite groups called CSQ-normality of subgroups. Using this subgroup property, we determine the structure of finite groups with some CSQ-normal subgroups of Sylow subgroups. As an application of our results, some recent results are generalized.

A note on solvable vertex stabilizers of s -transitive graphs of prime valency

Song-Tao Guo;  Hailong Hou;  Yong Xu — 2015

Czechoslovak Mathematical Journal

A graph X , with a group G of automorphisms of X , is said to be ( G , s ) -transitive, for some s ≥ 1 , if G is transitive on s -arcs but not on ( s + 1 ) -arcs. Let X be a connected ( G , s ) -transitive graph of prime valency p ≥ 5 , and G v the vertex stabilizer of a vertex v ∈ V ( X ) . Suppose that G v is solvable. Weiss (1974) proved that | G v | ∣ p ( p - 1 ) 2 . In this paper, we prove that G v ≅ ( ℤ p ⋊ ℤ m ) × ℤ n for some positive integers m and n such that n div m and m ∣ p - 1 .

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