Non-Solvable Graph of a Finite Group and Solvabilizers

2013/07/10 by Doron Hai-Reuven, Hai-Reuven, Doron · 2 citations
Mathematics · Neuroscience · #Finite Group Theory Research #Rings, Modules, and Algebras #Nuclear Receptors and Signaling

paper · pdf · doi:10.48550/arxiv.1307.2924

Abstract

Let G be a finite group. For x ∈ G, we define the solvabilizer of x in G, denoted sol_G(x), to be the set \g ∈ G | ⟨ g,x ⟩ is solvable\. A group G is an S-group if sol_G(x) is a subgroup of G for every x ∈ G. In this paper we prove that G is solvable ⇔ G is an S-group. Secondly, we define the non-solvable graph of G (denoted \mathcal S_G). Its vertices are G and there is an edge between x,y ∈ G whenever ⟨ x,y ⟩ is not solvable. If S(G) is the solvable radical of G and G is not solvable, we look at the induced graph over G ∖ S(G), denoted \mathcal S_G. We prove that if G is not solvable, then \mathcal S_G is irregular. In addition, we prove some properties of solvabilizers and non-solvable graphs.

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• The solvable Graph of a finite-dimensional Lie Algebra
• A note on solvable graphs of finite groups

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