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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2401.00767 |
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| _version_ | 1866912146912182272 |
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| author | Zhang, Shengmin |
| author_facet | Zhang, Shengmin |
| contents | Let $G$ be a finite group, and $g \in G$. Then $g$ is said to be a vanishing element of $G$, if there exists an irreducible character $χ$ of $G$ such that $χ(g)=0$. Denote by ${\rm Vo} (G)$ the set of the orders of vanishing elements of $G$. We say a non-abelian group $G$ is V-recognizable, if any group $N$ with ${\rm Vo} (N) = {\rm Vo} (G)$ is isomorphic to $G$. In this paper, we investigate the V-recognizability of $E_8 (p)$, where $p$ is a prime number. As an application, among the 610 primes $p$ with $p<10000$ and $p \equiv 0,1,4\,(\!\!\!\mod 5)$, we obtain that the method is always valid for confirming the V-recognizability of $E_8 (p)$ for all such $p$ but $ 919,1289,1931,3911,4691,5381$ and $7589 $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00767 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A new characterization of $E_8 (p)$ via its vanishing elements Zhang, Shengmin Group Theory 20C15 Let $G$ be a finite group, and $g \in G$. Then $g$ is said to be a vanishing element of $G$, if there exists an irreducible character $χ$ of $G$ such that $χ(g)=0$. Denote by ${\rm Vo} (G)$ the set of the orders of vanishing elements of $G$. We say a non-abelian group $G$ is V-recognizable, if any group $N$ with ${\rm Vo} (N) = {\rm Vo} (G)$ is isomorphic to $G$. In this paper, we investigate the V-recognizability of $E_8 (p)$, where $p$ is a prime number. As an application, among the 610 primes $p$ with $p<10000$ and $p \equiv 0,1,4\,(\!\!\!\mod 5)$, we obtain that the method is always valid for confirming the V-recognizability of $E_8 (p)$ for all such $p$ but $ 919,1289,1931,3911,4691,5381$ and $7589 $. |
| title | A new characterization of $E_8 (p)$ via its vanishing elements |
| topic | Group Theory 20C15 |
| url | https://arxiv.org/abs/2401.00767 |