奇异元个数对线性群PSL(2,p)的刻画

Characterization of PSL(2,p)by the Number of Singular Elements

设G是有限群,π(G)表示|G|的所有素因子的集合.若r∈π(G),r-奇异元g∈G表示g的元阶能被r整除,并设μ_(r)(G)为r-奇异元在G中的比例.令μ(G)={μ_(r)(G):r∈π(G)},利用μ(G)刻画了线性群PSL(2,p),并证明了:若μ(G)=μ(PSL(2,p)),则G?PSL(2,p),这里p是素数.

Given a finite group G,let π(G) represent the set of all primes in |G|.Given r∈π(G),g∈G is r-singular element if its order is divisible by r.Denote μ_(r)(G) the probability of r-singular elements in G.In this paper,we define μ(G)={μ_(r)(G):r∈π(G)} and use μ(G) to characterize linear groups PSL(2,p),where p is a prime.Moreover,we obtain that if μ(G)=μ(PSL(2,p)),then G?PSL(2,p).