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

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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