On Sums of Sylow Numbers of Finite Groups

Abstract

Let G be a finite group. Let $$n_{p}(G)$$ be the number of Sylow p subgroup of G and $$\pi (G)$$ be the set of prime divisors of |G|. We set $$S(G)=\{p\in \pi (G)|n_{p}(G)>1\}$$ and $$\delta (G)=\sum _{p\in \pi (G)}n_{p}(G)$$ , and $$ \delta _{0}(G)=\sum _{p\in S(G)}n_{p}(G)$$ . In this paper, we study groups G with small $$\delta (G)$$ and $$\delta _{0}(G)$$ . Furthermore, we will show that if G is a non-solvable group with $$C_{G}(N)=\{1\}$$ , where the minimal normal subgroup N of G is the last member of the derived series of G, then $$|G:G^{^{\prime }}|<\delta _{0}(G)$$ .