作者yusd24 (阿乡)
看板Math
标题Re: [代数] 一题考古题
时间Thu Feb 2 14:21:07 2012
※ 引述《tasukuchiyan (Tasuku)》之铭言：
: Let N be a normal subgroup of a finite group G. Suppose that |N|=5 and
: |G| is odd. Prove that N is contained in the center of G.
: 不知道该如何下手，请问有任何想法吗？谢谢。
|N|=5, so N is cyclic. Let g be its generator.
For any h in G, we must show that hgh^{-1}=g.
Since N is normal, we know hgh^{-1}=g^i, i=1,2,3,4.
Let o(h)=m. Note that m is an odd number.
g=h^mgh^{-m}=h^{m-1}hgh^{-1}h^{-m+1}=h^{m-1}g^ih^{-m+1}
=.......................=g^{i^m}
So we have i^m-1=0 (mod 5), with m odd. By Fermat's theorem, i^4=1 (mod 5)
we may assume m=1 or 3 (m odd iff m-4k odd)
In any case, i=1 is the only solution. This impies hgh^{-1}=g^i=g.
N is contained in C(G).
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1F：推 tasukuchiyan:感谢 02/02 16:12