A new characterization of the projective linear groups by the Sylow numbers

Abstract

Let G be a finite group, pi (G) be the set of primes p such that G contains an element of order p and n_{p}(G) be the number of Sylow p-subgroup of G, that is, n_{p}(G)=|Syl_{p}(G)|. Set NS(G):=\{n_{p}|p\in \pi (G)\}, the set of the all of the number of Sylow subgroups of G. In this paper, we show that the linear groups PSL(2, q) are recognizable by NS(G) and order. Also we prove that if NS(G)=NS(PSL(2,8)$), then G is isomorphic to PSL(2,8) or Aut(PSL(2,8)).