Re: [代數] 一題考古題
※ 引述《tasukuchiyan (Tasuku)》之銘言：
: 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.
: 不知道該如何下手，請問有任何想法嗎？謝謝。
G acts on N=<g> by conjugation.
Consider the orbits decomposition of N.
it's easily to show that |G．g^0|=1 and
|G．g|=|G．g^i| for i=1~4. (use Z/5 is a field)
So the orbits formula reads: 5=1+(# of non trivial orbits)|G．g|.
Hence #(G．g)|4.
If G．g={g}, then N is in Z(G), as <g>=N.
If |G．g| =2,4, then |G．g| = |G|/|G_g| is odd as |G| is odd. ><.
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