The first goal of this paper is to prove that if $ G $ is a finite noncyclic simple group and $ p $ is a prime divisor of $ |G| $ ;then $ n_{p}(G)\geq|\operatorname{Syl}_{p}(G)|+1 $ , where $ |\operatorname{Syl}_{p}(G)| $ is the size of a Sylow p-subgroup of $ G $ .The invariant $ \gamma(G) $ is a definite integral of the Sylow polynomial of a finite group $ G $ .Our second goal is to show that $ A_{5} $ is the only finite simple group $ G $ with $ \gamma(G)=\frac{9}{2} $ .
