كليدواژه :
non , solvable graph , sovabilizer , finite grou
چكيده فارسي :
The non-solvable graph of a finite group $G$, denoted by
${\cal S}_{G}$, is a simple graph whose vertices are the elements
of $G$ and there is an edge between two elements $x, y\in G$ if and only if
$\langle x, y\rangle$ is not solvable. If $R$ is the solvable
radical of $G$, the isolated vertices in ${\cal S}_{G}$ are
exactly the elements of $R$. Thus, in the case when $G$ is a
non-solvable group, it is wise to consider the
induced subgraph over $G\setminus R$ which is denoted by
$\widehat{{\cal S}_G}$. Let $G$ be a finite group and $x\in G$.
The solvabilizer of $x$ with respect to $G$, denoted by
$Sol_G(x)$, is the set $\{y\in G\ |\ \langle x, y\rangle \ {\rm
is\ solvable}\}$. In this paper, we are going to study some properties of
$\widehat{{\cal S}_G}$ and the structure of $Sol_G(x)$ for every
$x\in G$, more precisely.
