Some results on Sylow numbers of finite groups

We study the group structure in terms of the number of Sylow p-subgroups, which is denoted by n_p(G). The first part is on the group structure of finite group G such that n_p(G) = n_p(G/N), where N is a normal subgroup of G. The second part is on the average Sylow number asn(G) and we prove that if G is a finite nonsolvable group with asn(G) < 39/4 and asn(G) ≠ 29/4, then G/F(G) ≌ A₅, where F(G) is the Fitting subgroup of G. In the third part, we study the nonsolvable group with small sum of Sylow numbers.