Counting conjugacy classes of subgroups of \rm PSL₂(p)

2024/11/04 by Jones, Gareth A.
#20D60 #20E32 #20G40 #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #Primary 20D99 #secondary 11N32

paper · doi:10.48550/arxiv.2411.02219

Abstract

We obtain formulae for the numbers of isomorphism and conjugacy classes of non-identity proper subgroups of the groups G=\rm PSL₂(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.

Related

• Finite groups with many p-regular conjugacy classes
• Groups of prime degree and the Bateman-Horn Conjecture
• Conjugacy classes of groups of prime order in\n \PGL_k+1(\ℂ)
• p-Regular conjugacy classes and p-rational irreducible characters
• Fixed conjugacy classes of normal subgroups and the k(GV)-problem