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Peer-reviewedCC BYJournal of Computational Algebra

Criteria for classifying prime graphs of PSL(2,

  • Thomas Michael Keller1
  • Zachary Martin2
  • Alexa Renner3
  • Gabriel Roca4
  • Eric Yu5
5 affiliations
  1. 1Department of Mathematics, Texas State University, 601 University Drive, San Marcos, TX, 78666, USA
  2. 2Department of Mathematics, Willamette University, 900 State St., Salem, OR 97301, USA
  3. 3Department of Mathematics, Rose-Hulman Institute of Technology, 5500 Wabash Ave., Terre Haute, IN 47803, USA; Corresponding author.
  4. 4Department of Mathematics, University of Central Florida, 4000 Central Florida Blvd, Orlando, FL 32816, USA
  5. 5Department of Mathematics, University of Pennsylvania, 209 S 33rd St, Philadelphia, PA 19104, USA
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Abstract

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 = PSL ( 2 , q ) for q ≠ 2 5 and | 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 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 PSL ( 2 , 2 f ) such that f ≥ 5 , f is prime, and | 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.

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Cite

BibTeXRIS
BibTeX
@article{keller2026criteria,
  title = {{Criteria for classifying prime graphs of PSL(2,}},
  author = {Thomas Michael Keller and Zachary Martin and Alexa Renner and Gabriel Roca and Eric Yu},
  journal = {Journal of Computational Algebra},
  year = {2026},
  month = oct,
  doi = {10.1016/j.jaca.2026.100051},
  url = {https://doi.org/10.1016/j.jaca.2026.100051},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Criteria for classifying prime graphs of PSL(2,
AU  - Thomas Michael Keller
AU  - Zachary Martin
AU  - Alexa Renner
AU  - Gabriel Roca
AU  - Eric Yu
PY  - 2026
DA  - 2026/10/30
JO  - Journal of Computational Algebra
DO  - 10.1016/j.jaca.2026.100051
UR  - https://doi.org/10.1016/j.jaca.2026.100051
AB  - 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 = PSL ( 2 , q ) for q ≠ 2 5 and | 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 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 PSL ( 2 , 2 f ) such that f ≥ 5 , f is prime, and | 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.
SN  - 2772-8277
ER  -