Finite groups with exactly 2p + 1 subgroups of order p

Let m_p(G) be the number of elements of order p in a finite group G, p prime. We prove that there is no finite simple group G with m_p(G) = 2p^2-p-1, and that a finite group with this property exists if and only if 2p+1 is prime or 3^f with f an odd prime. This concerns an open problem and extends the classification of finite groups with few elements of prime order.