Criteria for Classifying Prime Graphs of PSL(2, q)-Solvable Groups
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| Format: | Preprint |
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2025
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| _version_ | 1866909869880115200 |
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| author | Keller, Thomas Michael Martin, Zachary Renner, Alexa Roca, Gabriel Yu, Eric |
| author_facet | Keller, Thomas Michael Martin, Zachary Renner, Alexa Roca, Gabriel Yu, Eric |
| contents | For a finite group $G$, the prime graph $Γ(G)$ (also known as Gruenberg-Kegel graph) is defined to be the graph where the vertices are the primes that divide $|G|$ such that two vertices $p$ and $q$ share an edge if and only if there is an element of order $pq$ in $G$. The prime graphs of solvable groups have been classified. The prime graphs of groups whose noncyclic composition factors are isomorphic to a single nonabelian simple group $T$ where $|T|$ is divisible by three or four distinct primes have been classified except for the cases where $T = \operatorname{PSL}(2,q)$ for $q\neq 2^5$ and $|\operatorname{PSL}(2,q)|$ is divisible by exactly four primes.
In this paper, we provide criteria for general classification results for certain classes of $T$, and then use them to classify the prime graphs of some $T$-solvable groups for $T$ a suitably small $\operatorname{PSL}(2, q)$-group. We also provide general results on the prime graphs of $T$-solvable groups where $T$ is a member of the possibly infinite family of groups $\operatorname{PSL}(2, 2^f)$ such that $f\geq 5, f$ is prime, and $|\operatorname{PSL}(2, 2^f)|$ is divisible by exactly four primes. This is the first paper to prove general results about the prime graphs of $T$-solvable groups where $T$ belongs to a large (probably infinite) family of groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_21979 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Criteria for Classifying Prime Graphs of PSL(2, q)-Solvable Groups Keller, Thomas Michael Martin, Zachary Renner, Alexa Roca, Gabriel Yu, Eric Group Theory 20D60 and 05C25 For a finite group $G$, the prime graph $Γ(G)$ (also known as Gruenberg-Kegel graph) is defined to be the graph where the vertices are the primes that divide $|G|$ such that two vertices $p$ and $q$ share an edge if and only if there is an element of order $pq$ in $G$. The prime graphs of solvable groups have been classified. The prime graphs of groups whose noncyclic composition factors are isomorphic to a single nonabelian simple group $T$ where $|T|$ is divisible by three or four distinct primes have been classified except for the cases where $T = \operatorname{PSL}(2,q)$ for $q\neq 2^5$ and $|\operatorname{PSL}(2,q)|$ is divisible by exactly four primes. In this paper, we provide criteria for general classification results for certain classes of $T$, and then use them to classify the prime graphs of some $T$-solvable groups for $T$ a suitably small $\operatorname{PSL}(2, q)$-group. We also provide general results on the prime graphs of $T$-solvable groups where $T$ is a member of the possibly infinite family of groups $\operatorname{PSL}(2, 2^f)$ such that $f\geq 5, f$ is prime, and $|\operatorname{PSL}(2, 2^f)|$ is divisible by exactly four primes. This is the first paper to prove general results about the prime graphs of $T$-solvable groups where $T$ belongs to a large (probably infinite) family of groups. |
| title | Criteria for Classifying Prime Graphs of PSL(2, q)-Solvable Groups |
| topic | Group Theory 20D60 and 05C25 |
| url | https://arxiv.org/abs/2510.21979 |