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 is a finite group,define the number of elements in of order Give an example of two nonisomorphic not necessarily abelian finite groups and for which for all integers .
Hint.Consider of order
2.(P273 Lemma5.40)There is no nonabelian simple group of order where is prime,
3.(P276 Theorem5.47)If is a prime and then the unitriangular group is a Sylow subgroup of
A corollary:a finite group can be imbedded in
4.(P278 Exercise5.30)Prove that there is no simple group of order 120.
Hint.If a simple group have a proper subgroup of index , then can be imbedded in
5.(P287 Exercise5.33)Let be a prime and let be a nonabelian group of order Prove that
Hint.Notice that and is abelian.
6.(P288 Exercise5.45)Let be a group containing elements and such that the orders of and are pairwise relatively prime; prove that is not solvable.
Hint.Consider the quotient group of which is abelian, and we will find that .
7.(P295 Proposition5.67)If is a simple group of order 60,then
Hint.Only have to prove that has a subgroup of index 5.
8.(P310 Exercise5.61)If is a finitely generated group and is a positive integer, prove that has only finitely many subgroups of index .
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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