Independence and strong independence complexes of finite groups
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910270946803712 |
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| author | Lucchini, Andrea Stanojkovski, Mima |
| author_facet | Lucchini, Andrea Stanojkovski, Mima |
| contents | Let $G$ be a finite group. In 2024, Cameron introduced two different concepts of independence (namely independence and strong independence) for the subsets of $G$, yielding to the definition of two simplicial complexes whose vertices are the elements of $G$. The strong independence complex $\tildeΣ(G)$ turns out to be a subcomplex of the independence complex $Σ(G)$. We discuss several invariant properties related to these complexes and ask a number of questions inspired by our results and the examples we construct. We study then the particular case of complexes on finite abelian groups, giving a characterization of the finite groups realizing them. In conclusion, answering a question of Cameron, we classify all finite groups in which the two concepts of independence coincide. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19778 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Independence and strong independence complexes of finite groups Lucchini, Andrea Stanojkovski, Mima Group Theory Let $G$ be a finite group. In 2024, Cameron introduced two different concepts of independence (namely independence and strong independence) for the subsets of $G$, yielding to the definition of two simplicial complexes whose vertices are the elements of $G$. The strong independence complex $\tildeΣ(G)$ turns out to be a subcomplex of the independence complex $Σ(G)$. We discuss several invariant properties related to these complexes and ask a number of questions inspired by our results and the examples we construct. We study then the particular case of complexes on finite abelian groups, giving a characterization of the finite groups realizing them. In conclusion, answering a question of Cameron, we classify all finite groups in which the two concepts of independence coincide. |
| title | Independence and strong independence complexes of finite groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2503.19778 |