13年《基础代数》复习题

《基础代数》复习题

0.概念：群中元素的阶数；正规子群；商群；单群；（左、右）理想；商环；分式环；整环；环的特征；模；域；代数元；1．求二面体群的全部子群、正规子群。

2．（1）写出所有不同构的18、36阶交换群。

（2）写出所有不同构的p2阶群, p为奇素数。

3．（1）证明56阶群有正规的Sylow 2-子群或者有正规的Sylow 7-子群。

（2）证明p2q阶群不是单群。

（3）设是p, q是不同的素数，证明pq阶群都有正规的Sylow 子群.

4. 证明任意2p阶群都同构于循环群或者二面体群。

5．判断下面的命题是否正确？对正确的请加以证明；对不正确的请举出反例说明。

(1)在有限群中，如果a与b共轭，c与d共轭，那么ac

与bd共轭。

(2)如果H是G的正规子群，K是H的正规子群，那么K

是G的正规子群。

(3)设Z(G)是有限群G的中心，并且G/Z(G)是循环群，那

么G是交换群。

(4) 设G是有限群，那么对它的阶数|G|的每个因子n，G都有n阶子群。

(5) 设G是有限群，G的任意指数为2、3的子群都是G的正规子群。

6．用GL(n,q) 和SL(n,q)分别表示有限域GF(q)上n维向量空间上全体可逆线性变换、行列式为1的全体可逆线性变换所构成的群.。

（1）分别求GL(n,q) 和SL(n,q)的阶数。

（2）分别求GL(n,q) 和SL(n,q)的中心。

7．设M2(F)是域F上全体2级矩阵按矩阵的加法、乘法所构成的环。(1) 求M2(F)的所有左理想和右理想。

(2)求M2(F)的所有理想。

8．设G是有限群，p是其阶数|G|的最小素因子，证明G的任意指数为p的子群都是G的正规子群。

9．设G是有限群，如果Aut G = 1，那么G的阶数为1或2。

10. 求5次交错群、4次对称群的所有不共轭的子群.

11 叙述群同态基本定理、Sylow定理、同构定理.

12. 试给出G的子群H是正规子群的几个等价条件.

13 求在模18剩余类环Z18中的所有零因子、幂零元.

14设G是有限群，p是其阶数|G|的最小素因子，证明G的任意阶数为p的正规子群包含在G的中心中。

15 设a是有限域F=GF(2)上多项式31

++的根，

x x

（1）求扩域F(a)作为有限域F上线性空间的一组基；

（2）化简432

+++++Section A

(1)(1)

a a a a a

之所以不把二氧化碳列为污染物，是因为二氧化碳是大气的天然成份，植物进行生物合成需要二氧化碳。正如人们普遍认为人的饮食不可缺少维生素D一样，也都认为二氧化碳是大气不可缺少的成份。但是摄入过多的维生素D会有很大的副作用。生命系统，不论是生态系统还是生物，都需要在某些化合物之间保持微妙的平衡，以确保系统功能正常。尽管少量的某种物质是必要的，但当该物质过量出现威胁生态系统的健康时，就具有了毒副作用。

1．第一句的表语从句中有it is a natural component of the atmosphere and needed by plants to carry out biological synthesis，其中it is a...and needed...可理解为needed前面省略了it is，后面的不定式充当目的状语，意义相当于plants need carbon dioxide to carry out biological synthesis。

2．第二句是该段落中最难理解的句子，难就难在如何理解No one would argue that...any more than one would argue...，字面意思是“一个人为…争辩的力度不会超过…，意译就是“对两者相信的程度相同”。3．第四句。Living systems, be they an ecosystem or an organism...就是“不论是生态系统，还是生物(有机体)”。require后的从句是虚拟语气，动词用原形。excess可当形容词用，相当于excessive。名词quantity number和amount可与介词in连用充当状语，如import oil in large quantities(大量进口石油)或kill the enemy in large numbers(消灭大量)。The fast development of telecommunications has made the cell (mobile)

phone popular among (a favorite of) high schoolers (high school students). Despite its multiple (many) functions, the cell phone has some negative (adverse) effect on the academic performance of teenagers，such as cheating on a test or distraction (lack of attention/lack of concentration/a shortened attention span) in class. Besides, regular use of a cell phone ends up with less face-to-face communication while chronic exposure to the radiation from the cell phone can lead to (result in/cause) the decline of memory or increase the chance (odds/prospect) of developing a brain tumor.

1．翻译“电脑”或“手机”时要注意冠词的使用，不能写成use computer。2．经常用academic来表达与“学习”相关的意义，如academic excellence(学习好)，academic performance(学习成绩)。

3．能使用名词的时候尽量不要用不定式或动名词。例如，第三句中的“经常使用”最好不要翻译为often using或if we use...very often...，同样，“长期接触”不要翻译成if we are exposed to...for a long time，这样的表达不简练。请看下列汉语词组的翻译：

……有助于全球范围内使用电脑…contribute to worldwide use of the computer

避免接触有关暴力的图书avoid exposure to books about violence

经常运动能……regular exercise can...未能找到解决方案表

明……failure to find any solutions suggests...

尽管我坚信……despite my conviction that...

我最惊讶的是……the biggest surprise to me is...

4．“接触”不好翻译。有人把“我经常接触老外”翻译成I often touch foreigners；某电视台把电视节目“非常接触”译为unique touch，不知用什么touch ? touch通常指“用手触摸”或“联系”，如stay in touch with former teachers或lose touch with reality。be exposed to常见的意义是“使……遭受”，通常有“受到负面影响”的含义，如be exposed to danger/attack/ridicule/pollutants，但也可说be exposed to classical music/Western culture/books on history，介词必须是to。contact有时可代替这两个词，通常意义有：

have close contact with various patients近距离接触各类患者

eye contact目光接触body contact身体接触

come into contact with new ideas接触新思想

any contact with this liquid will...只要接触这种液体

have wide contact with these politicians广泛接触这些政客(打交道) 5．“增加……的概率”往往用chance odds或prospect翻译，尤其是chance，但接of doing...，而不接to do...。

没人会花很长时间来决定是阅读印刷的广告还是听播放的广告。如果广告不能马上吸引人们的注意力，使其专注片刻，足以理解广告内容，那么这个广告会惨败。因此，广告词的效果必须是立竿见影，马上让你心动。广告词必须让人关注所介绍的产品，突出其性能，简明介绍购买它的理由，最好使读者或听众对产品的介绍留下深刻印象，并能长久回荡在其脑海里。

Jobs come easily to college graduates who …( Some college graduates can find jobs / work / employment / get employed easily, who…) are characterized by academic excellence and the willingness to do low-income / poorly-paid jobs. The two factors are equally important / of equal importance / equal in importance. Working where you can prove your own worth / value is a crucial / key step towards success. For example, Obama started as a community organizer, who was unlikely to expect (could not have expected / probably did not expect ) that he could becomepresident of the United States.

以下翻译方式酌情扣分：

1. …graduates are easy to find jobs. 属于严重错误。

2. …find a work. work 不可数.

3. ….characterized by willing to…或…willingness of doing….

4. …factors are the same important. 属于严重错误。

5. …prove the value of oneself / oneself’s value (worth)

6. …who was impossible to expect…属于严重错误。…who could not expect…

1.1 Jobs come easily to college graduates who are characterized by academic excellence and thewillingness to do low-income jobs.

1.2 Some college graduates can get employed easily. What they have in common is good academic performance and willingness to do low-income jobs. 1.3 It is easy for some college graduates to find a job. They usually have a good academic record. They are also willing to do poorly-paid jobs.

2.1 The two factors are of equal importance. equal in importance

2.2 The two factors are equally important

3.1 It is a key step to success to be able to work in a position that shows your abilities. 3.2 Working where you can prove your own va lue is a crucial step towards success.

4.1 For example, Obama started as a community organizer, who was unlikely to expect that he could become president of the United States.

4.2 For example, Obama started as a community organizer. He could not have expected that he could become president of the United States. 4.3 For example, Obama’s first job was a community organizer. It was impossible for him to anticipate presidency of the United States.