Non-Solvable Graph of a Finite Group and Solvabilizers

Abstract

Let $G$ be a finite group. For $x \in G$, we define the solvabilizer of $x$ in $G$, denoted $sol_G(x)$, to be the set $\{g \in G \mid \langle g,x \rangle$ is solvable$\}$. A group $G$ is an S-group if $sol_G(x)$ is a subgroup of $G$ for every $x \in G$. In this paper we prove that $G$ is solvable $\Leftrightarrow$ $G$ 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 $x,y \in G$ whenever $\langle x,y \rangle$ is not solvable. If $S(G)$ is the solvable radical of $G$ and $G$ is not solvable, we look at the induced graph over $G \setminus S(G)$, denoted $\hat{{\mathcal S}_{G}}$. We prove that if $G$ is not solvable, then $\hat{{\mathcal S}_{G}}$ is irregular. In addition, we prove some properties of solvabilizers and non-solvable graphs.