Normal Subgroup Based Power Graph of a finite Group

Abstract

For a finite group $G$ with a normal subgroup $H$, the normal subgroup based power graph of $G$, denoted by ${\Gamma }_H(G)$ whose vertex set $V({\Gamma }_H(G))=(G\setminus H)\bigcup \{e\}$ and two vertices $a$ and $b$ are edge connected if $aH=b^{m}H$ or $bH=a^{n}H$ for some $m,n\in \mathbb{N}$. In this paper we obtain some fundamental characterizations of the normal subgroup based power graph. We show some relation between the graph ${\Gamma }_H(G)$ and the power graph $\Gamma (\frac{G}{H})$. We show that ${\Gamma }_H(G)$ is complete if and only of $\frac{G}{H}$ is cyclic group of order $1$ or $p^m$, where $p$ is prime number and $m\in \mathbb{N}$. ${\Gamma }_H(G)$ is planar if and only if $|H|=2$ or $3$ and $\frac{G}{H}\cong {\mathbb{Z}}_2\times {\mathbb{Z}}_2\times \cdots \times {\mathbb{Z}}_2$. Also ${\Gamma }_H(G)$ is Eulerian if and only if $|G|\equiv |H|$ mod$2$.

14 pages, 4 figures