Counting conjugacy classes of subgroups of PSL₂(p)

Abstract

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

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