Quantitive characterization of simple group PSL(2,p)

This paper establishes the following two results. Let p\geq 5 be a prime number, if the order of a simple group G is \dfrac(p+1)p(p-1)2 , then G\cong PSL(2,p) . If the order of the group G is \dfrac(p+1)p(p-1)2 , and the maximal order of elements in G is p , then either G\cong PSL(2,p) or p=7 and G\cong \mathrmZ_2^3\colon (\mathrmZ_7\colon \mathrmZ_3) . The latter result is a main finding of Bull. Korean Math. Soc., 2015, 52(6): 2025-2034 and we present a completely different method for its proof. The proofs in this paper utilize elementary group theory and basic knowledge of permutation groups, without relying on the classification theorem of finite simple groups.