On characterization of groups by isomorphism type of Gruenberg-Kegel graph
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910915313532928 |
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| author | Chen, Mingzhu Maslova, Natalia V. Zinov'eva, Marianna R. |
| author_facet | Chen, Mingzhu Maslova, Natalia V. Zinov'eva, Marianna R. |
| contents | The Gruenberg-Kegel graph (or the prime graph) $Γ(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of order $rs$ in $G$. A group $G$ is recognizable by isomorphism type of Gruenberg--Kegel graph if for every group $H$ the isomorphism between $Γ(H)$ and $Γ(G)$ as abstract graphs (i.\,e. unlabeled graphs) implies that $G\cong H$. In this paper, we prove that finite simple exceptional groups of Lie type ${^2}E_6(2)$ and $E_8(q)$ for $q \in \{3, 4, 5, 7, 8, 9, 17\}$ are recognizable by isomorphism type of Gruenberg-Kegel graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14703 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On characterization of groups by isomorphism type of Gruenberg-Kegel graph Chen, Mingzhu Maslova, Natalia V. Zinov'eva, Marianna R. Group Theory 20D60, 20D06, 05C99 The Gruenberg-Kegel graph (or the prime graph) $Γ(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of order $rs$ in $G$. A group $G$ is recognizable by isomorphism type of Gruenberg--Kegel graph if for every group $H$ the isomorphism between $Γ(H)$ and $Γ(G)$ as abstract graphs (i.\,e. unlabeled graphs) implies that $G\cong H$. In this paper, we prove that finite simple exceptional groups of Lie type ${^2}E_6(2)$ and $E_8(q)$ for $q \in \{3, 4, 5, 7, 8, 9, 17\}$ are recognizable by isomorphism type of Gruenberg-Kegel graph. |
| title | On characterization of groups by isomorphism type of Gruenberg-Kegel graph |
| topic | Group Theory 20D60, 20D06, 05C99 |
| url | https://arxiv.org/abs/2504.14703 |