Difference graphs of finite abelian groups with two Sylow subgroups
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916227488677888 |
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| author | Bošnjak, Ivica Madarász, Rozália Zahirović, Samir |
| author_facet | Bošnjak, Ivica Madarász, Rozália Zahirović, Samir |
| contents | The power graph and the enhanced power graph of a group $\mathbf G$ are simple graphs with vertex set $G$; two elements of $G$ are adjacent in the power graph if one of them is a power of the other, and they are adjacent in the enhanced power graph if they generate a cyclic subgroup. The difference graph of a group $\mathbf G$, denoted by $\mathcal D(\mathbf G)$, is the difference of the enhanced power graph and the power graph of group $\mathbf G$ with all the isolated vertices removed. In this paper, we prove that, if a pair of finite abelian groups of order divisible by at most two primes have isomorphic difference graphs, then they are isomorphic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18167 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Difference graphs of finite abelian groups with two Sylow subgroups Bošnjak, Ivica Madarász, Rozália Zahirović, Samir Combinatorics Group Theory 05C25, 20D60 The power graph and the enhanced power graph of a group $\mathbf G$ are simple graphs with vertex set $G$; two elements of $G$ are adjacent in the power graph if one of them is a power of the other, and they are adjacent in the enhanced power graph if they generate a cyclic subgroup. The difference graph of a group $\mathbf G$, denoted by $\mathcal D(\mathbf G)$, is the difference of the enhanced power graph and the power graph of group $\mathbf G$ with all the isolated vertices removed. In this paper, we prove that, if a pair of finite abelian groups of order divisible by at most two primes have isomorphic difference graphs, then they are isomorphic. |
| title | Difference graphs of finite abelian groups with two Sylow subgroups |
| topic | Combinatorics Group Theory 05C25, 20D60 |
| url | https://arxiv.org/abs/2404.18167 |