### On Normal Graph of a Finite Group

#### Abstract

Suppose $G$ is a finite group and $\mathcal{C}(G)$ denotes the

set of all conjugacy classes of $G$. The normal graph of $G$,

$\mathcal{N}(G)$, is a finite simple graph such that

$V(\mathcal{N}(G)) = \mathcal{C}(G)$. Two conjugacy classes $A$

and $B$ in $\mathcal{C}(G)$ are adjacent if and only if there is

a proper normal subgroup $N$ such that $A \cup B \leq N$. The

aim of this paper is to study the normal graph of a finite group

$G$. It is proved, among others, that the groups with identical character table have isomorphic normal graphs. The normal graphs of some classes of finite groups are also obtained and some open questions are posed.

set of all conjugacy classes of $G$. The normal graph of $G$,

$\mathcal{N}(G)$, is a finite simple graph such that

$V(\mathcal{N}(G)) = \mathcal{C}(G)$. Two conjugacy classes $A$

and $B$ in $\mathcal{C}(G)$ are adjacent if and only if there is

a proper normal subgroup $N$ such that $A \cup B \leq N$. The

aim of this paper is to study the normal graph of a finite group

$G$. It is proved, among others, that the groups with identical character table have isomorphic normal graphs. The normal graphs of some classes of finite groups are also obtained and some open questions are posed.

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