Counting conjugacy classes of subgroups of ${\rm PSL}_2(p)$

Document Type : Research Paper

Author

School of Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, UK

10.22108/ijgt.2025.144154.1942

Abstract

This work is motivated by results obtained and problems posed by Bianchi, Camina, Lewis, Pacifici and Sanus, counting conjugacy classes of non-self-normalising subgroups of finite groups. We obtain formulae for the numbers of isomorphism and conjugacy classes of non-identity proper subgroups of the groups $G={\rm PSL}_2(p)$, $p$ prime, and for the numbers of those conjugacy classes which do or do not consist of self-normalising subgroups. The formulae are used to prove lower bounds $17$, $18$, $6$ and $12$ respectively satisfied by these invariants for all $p>37$. A computer search carried out for a different but related problem shows that these bounds are attained for over a million primes $p$; we show that if the Bateman--Horn Conjecture is true, they are attained for infinitely many primes. Also, assuming no unproved conjectures, we use a result of Heath-Brown to obtain upper bounds for these invariants, valid for an infinite set of primes $p$.

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References

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International Journal of Group Theory
Volume 15, Issue 3 - Serial Number 3
September 2026
Pages 123-134

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History

  • Receive Date: 01 February 2025
  • Revise Date: 30 July 2025
  • Accept Date: 30 July 2025
  • Published Online: 30 September 2025

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