Genus and crosscap of Normal subgroup based power graphs of finite groups
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| Format: | Preprint |
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2024
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| _version_ | 1866917630680498176 |
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| author | Parveen Manisha Kumar, Jitender |
| author_facet | Parveen Manisha Kumar, Jitender |
| contents | Let $H$ be a normal subgroup of a group $G$. The normal subgroup based power graph $Γ_H(G)$ of $G$ is the simple undirected graph with vertex set $V(Γ_H(G))= (G\setminus H)\cup \{e\}$ and two distinct vertices $a$ and $b$ are adjacent if either $aH = b^m H$ or $bH=a^nH$ for some $m,n \in \mathbb{N}$. In this paper, we continue the study of normal subgroup based power graph and characterize all the pairs $(G,H)$, where $H$ is a non-trivial normal subgroup of $G$, such that the genus of $Γ_H(G)$ is at most $2$. Moreover, we determine all the subgroups $H$ and the quotient groups $\frac{G}{H}$ such that the cross-cap of $Γ_H(G)$ is at most three. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_03895 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Genus and crosscap of Normal subgroup based power graphs of finite groups Parveen Manisha Kumar, Jitender Combinatorics Group Theory 05C25 Let $H$ be a normal subgroup of a group $G$. The normal subgroup based power graph $Γ_H(G)$ of $G$ is the simple undirected graph with vertex set $V(Γ_H(G))= (G\setminus H)\cup \{e\}$ and two distinct vertices $a$ and $b$ are adjacent if either $aH = b^m H$ or $bH=a^nH$ for some $m,n \in \mathbb{N}$. In this paper, we continue the study of normal subgroup based power graph and characterize all the pairs $(G,H)$, where $H$ is a non-trivial normal subgroup of $G$, such that the genus of $Γ_H(G)$ is at most $2$. Moreover, we determine all the subgroups $H$ and the quotient groups $\frac{G}{H}$ such that the cross-cap of $Γ_H(G)$ is at most three. |
| title | Genus and crosscap of Normal subgroup based power graphs of finite groups |
| topic | Combinatorics Group Theory 05C25 |
| url | https://arxiv.org/abs/2404.03895 |