Let G be a finite group and let p be an odd prime. We denote by m_p(G) the number of elements of order p in G. If s_p(G) denotes the number of subgroups of order p, then m_p(G)=(p-1)s_p(G). Together with Frobenius' congruence s_p(G) ≡ 1(mod p), this shows that the first two possible positive values of m_p(G) are p-1 and p²-1.

We first investigate the first threshold and obtain a solvability criterion for finite groups satisfying 0 < m_p(G) < p²-1. We then study the boundary value m_p(G)=p²-1. In this case, the group has exactly p+1 subgroups of order p, and the conjugation action on these subgroups provides a natural permutation representation of degree p+1. This action will be the main tool in the analysis of the simple case. Finally, we prove that if S is a finite nonabelian simple group and p >= 5, then m_p(S)=p²-1 if and only if S ≅ PSL₂(p). We also show that no finite nonabelian simple group satisfies m₃(S)=8.

The author would like to thank the anonymous referee for the careful reading of the manuscript and for the constructive comments and insightful suggestions, which significantly improved the presentation and exposition of this paper.