Characterization Of Finite Groups By Orders

Let G be a finite group. K1(G) denote the largest element order of G. and K2(G). the second largest order, and K3(G), the third largest order. In this paper, we characterize some finite groups by using the order of G and K1(G)(For some special cases, we use K2(G) and K3(G)). The paper consists of7setions and the main results are in sections3、4、5、6.In Section3. we show that every sporadic simple group G can be uniquely determined by the order of G and K1(G).In Section4. we show that the automorphism group G of every sporadic simple group can be uniquely determined by the order of G and Ki(C), where i≤3.In Section5, we show that every alternating group An (n≤15) can be uniquely determined by the order of An and K1(An).In Section6. we show that symmetric group Sn (n≤15) can be uniquely deter-mined by the order of Sn and Ki(Sn), where i≤3.