Classifying prime graphs of finite groups -- a methodical approach
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866929415015890944 |
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| author | Keller, Thomas Michael Pettigrew, Gavin Solotko, Saskia Zheng, Lixin |
| author_facet | Keller, Thomas Michael Pettigrew, Gavin Solotko, Saskia Zheng, Lixin |
| contents | For a finite group $G$, the vertices of the prime graph $Γ(G)$ are the primes that divide $|G|$, and two vertices $p$ and $q$ are connected by an edge if and only if there is an element of order $pq$ in $G$. Prime graphs of solvable groups as well as groups whose noncyclic composition factors have order divisible by exactly three distinct primes have been classified in graph-theoretic terms. In this paper, we begin to develop a general theory on the existence of edges in the prime graph of an arbitrary $T$-solvable group, that is, a group whose composition factors are cyclic or isomorphic to a fixed nonabelian simple group $T$. We then apply these results to classify the prime graphs of $T$-solvable groups for, in a suitable sense, most $T$ such that $|T|$ has exactly four prime divisors. We find that these groups almost always have a 3-colorable prime graph complement containing few possible triangles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06595 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classifying prime graphs of finite groups -- a methodical approach Keller, Thomas Michael Pettigrew, Gavin Solotko, Saskia Zheng, Lixin Group Theory 20D60 (primary) 05C25 (secondary) For a finite group $G$, the vertices of the prime graph $Γ(G)$ are the primes that divide $|G|$, and two vertices $p$ and $q$ are connected by an edge if and only if there is an element of order $pq$ in $G$. Prime graphs of solvable groups as well as groups whose noncyclic composition factors have order divisible by exactly three distinct primes have been classified in graph-theoretic terms. In this paper, we begin to develop a general theory on the existence of edges in the prime graph of an arbitrary $T$-solvable group, that is, a group whose composition factors are cyclic or isomorphic to a fixed nonabelian simple group $T$. We then apply these results to classify the prime graphs of $T$-solvable groups for, in a suitable sense, most $T$ such that $|T|$ has exactly four prime divisors. We find that these groups almost always have a 3-colorable prime graph complement containing few possible triangles. |
| title | Classifying prime graphs of finite groups -- a methodical approach |
| topic | Group Theory 20D60 (primary) 05C25 (secondary) |
| url | https://arxiv.org/abs/2407.06595 |