The influence of the number of elements of prime order on the structure of finite groups

Let G be a finite group. We denote by n_{p}(G) the number of Sylow p-subgroup of G, that is, n_{p}(G)=|mathrm{Syl}_{p}(G)|. Denoted by m_{i}(G) the number of elements of order i of G. Given a positive integer n and a prime r, we write n_{r} to denote the full r-part of n, so we can factor n=n_{r}m, where m is not divisible by r. Now fix a prime p. We say that a positive integer n is a strong Sylow number for $p$ if for every prime q, the full q-part n_{q} of n satisfies n_{q}equiv 1 (mod p). Note that if n is a strong Sylow number for p, then nequiv 1 (mod p), and thus n is not divisible by p. Note also that the set of strong Sylow numbers for p is closed under multiplication. Let S be a nonabelian simple group that is not isomorphic to L_2 (r), where r is a Mersenne prime and let p be the greatest prime divisor of |S|. In [1, Conjecture E] A. Moreto conjectured that if a finite group G is generated by elements of order p and G has the same number of elements of order p as S, then G/(Z(G))≅S. In this paper, we verify the conjecture for the sporadic simple groups.