| As we know, for the nonabelian simple groups of order less than n, when n≤1000, n must be 60、168、360、504、660, and there are only five nonabelian simple groups of order less than n: A5 has order 60-. PSL(2,7) has order 168、As has order 360、PSL(2,8) has order 504 and PSL(2,11) has order 660. By using Sy-low theorem it’s easy to show that the simple group of order 60 is isomorphic to A5, [2], [3]. In [2] and [3], Huppert and Smith proved that the simple group of order 168 is isomorphic to PSL(2,7) by different elementary group methods, using the method in [6] to show that the simple group of order 660 is isomorphic to PSL(2,11) [2].In [4], Isaacs showed that the simple group of order 360 is isomorphic to A6 by character theory, it is a famous conclusion in finite group theory. In the books on group theory, there is complete proof about this conclusion neither in contexts nor exercises. For example, in the exercise 8.12[5], Rotman mean to show that the simple group of order 360 is isomorphic to A6 by elementary group method, but his hint "the simple group of order 360 has only 6 Sylow 5-subgroups" is wrong, since the number of Sylow 5-subgroups of A6 is 36. We discussed the phenomenon with Pro. Wujie Shi, Pro. Jiping Zhang, Pro. Caiheng Li, Pro. Ping Jin and other scholars. When we found the proof, we discussed with Pro. Jie Wang and Haoran Yu. Pro. Jie Wang showed the pure group theory proof of the conclusion of this article by Burnside transfer theorem in his notebook. Haoran Yu used the argument (Burnside transfer theorem)to prove the conclusion without character theory [4]. These works are also meaningful. Recently, Haoran Yu searched the references hard, and found the article of F.N.Cole[7]. Cole showed the conclusion of this article by using permutation group, in fact, the proof of this article is more elementary than [7] just by using Sylow theorem and basic permutation calculations, this paper is self-contained, the author hope this proof is meaningful for group theorists. |
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