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

TL;DR

This paper derives formulas for counting conjugacy classes of subgroups in PSL_2(p), establishes lower bounds for these counts, and discusses their behavior across primes, supported by computational evidence and conjectural assumptions.

Contribution

It provides explicit formulas for subgroup conjugacy class counts in PSL_2(p) and proves bounds that are attained infinitely often under certain conjectures.

Findings

Formulas for subgroup conjugacy class counts in PSL_2(p)

Lower bounds for these counts valid for all p > 37

Computational evidence shows bounds are attained for over a million primes

Abstract

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 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 $p$.