Rotman 上的一些題(4)

這一章是Rotman上的Groups II, 裡面像free group之類的以前沒怎麼接觸過。。。

1.(P268 Exercise5.14)If G is a finite group,define v_{k}(G)= the number of elements in G of order k. Give an example of two nonisomorphic not necessarily abelian finite groups G and G^{prime} for which v_{k}(G)=v_{k}(G^{prime}) for all integers k .

Hint.Consider G of order p^{3}.

2.(P273 Lemma5.40)There is no nonabelian simple group G of order |G|=p^{e}m, where p is prime, p
mid m,p^{e}
mid (m-1)!.

3.(P276 Theorem5.47)If p is a prime and q=p^{m}, then the unitriangular group mathrm{UT}(n,mathbb{F}_{q}) is a Sylow p- subgroup of mathrm{GL}(n,mathbb{F}_{q}).

A corollary:a finite p- group G can be imbedded in mathrm{UT}(|G|,mathbb{F}_{p}).

4.(P278 Exercise5.30)Prove that there is no simple group of order 120.

Hint.If a simple group G have a proper subgroup H of index n , then G can be imbedded in A_{n}.

5.(P287 Exercise5.33)Let p be a prime and let G be a nonabelian group of order p^{3}. Prove that Z(G)=G^{prime}.

Hint.Notice that |G|=p and G/Z(G) is abelian.

6.(P288 Exercise5.45)Let G be a group containing elements x and y such that the orders of x,y and xy are pairwise relatively prime; prove that G is not solvable.

Hint.Consider the quotient group G/H of G which is abelian, and we will find that x,yin H .

7.(P295 Proposition5.67)If G is a simple group of order 60,then Gcong A_{5}.

Hint.Only have to prove that G has a subgroup of index 5.

8.(P310 Exercise5.61)If G is a finitely generated group and n is a positive integer, prove that G has only finitely many subgroups of index n .

最後再補一個並不顯然的命題:存在兩個不同構的有限生成群,每一個都同構於另一個的一個子群。

到這裡也算是把學的很不紮實的近世代數重新過了一遍,Rotman後面的章節等以後複習代數學的時候再發吧,準備回家過年鹹魚一段時間了。

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