A solvability criterion for finite groups related to the number of Sylow subgroups

Let G be a finite group and let π(G) be the set of primes dividing the order of G. For each p∈π(G), the Sylow theorems state that the number of Sylow p-subgroups of G is equal to kp + 1 for some non-negative integer k. In this article, we characterize non-solvable groups G containing at most p2+1 Sylow p-subgroups for each p∈π(G). In particular, we show that each finite group G containing at most (p−1)p+1 Sylow p-subgroups for each p∈π(G) is solvable.