Rotman 上的一些題（4）

02-19

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。