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arXiv · 2411.02219

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

Gareth A. Jones
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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 $p$.

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Gareth A. Jones. 2024-11-04. Counting conjugacy classes of subgroups of ${\rm PSL}_2(p)$. https://arxiv.org/abs/2411.02219

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