推 turboho : 太感謝了！ 06/22 16:24
※ 引述《turboho (西卡拉)》之銘言：
: 最近作一個問題的時候遇到的東西，但是一直找不出證明或反例，丟上來給大家試試
: 問題：是否存在這樣的 group G 同時滿足下列條件:
: 1) G is nilpotent
: 2) G is finitely generated
: 3) G is infinite
: 4) Z(G) (G 的 center) is finite
: 我自己的猜測是不存在，即為
: Any finitely generated infinite nilpotent group has infinite center
: 不過我不會證 XD
的確是沒有這種G
upper central series 1=Z_0(G) < Z_1(G) < Z_2(G) < Z_3(G) < ... < Z_c(G)=G
Lemma: If Z_1(G) has exponent e, then all Z_{i+1}(G)/Z_i(G) has exponent
dividing e.
Proof: Enough to show i=1.
So let x in Z_2(G) and g in G. Then [x,g] in Z(G) and so
1=[x,g]^e=[x^e,g]
So x^e in Z(G), as desired. QED.
Corollary: Z(G) finite, G f.g. nilpotent => G finite.
Proof: From lemma, G has finite exponent.
So the abelianization G_ab is a finite abelian group.
Now each of the successive quotients of the lower central series is a
homomorphic image of a tensor power of G_ab, so are finite.
So G is finite. QED.
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『我思故我在』怎樣從法文變成拉丁文的：
je pense, donc je suis --- René Descartes, Discours de la Méthode (1637)
ego sum, ego existo --- ____, Meditationes de Prima Philosophia (1641)
ego cogito, ergo sum --- ____, Principia Philosophiae (1644)
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