课程名称︰代数导论一
课程性质︰数学系大二必修
课程教师︰李秋坤教授
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2019/11/8
考试时限（分钟）：180
试题 :
(满分100分)
(以下的属于符号都用ε代替)
1. (10%) Show that 〈(1,2),(1234)〉=S4. Is it true that〈(1,2),(1324)〉=S4?
2. (10%) Let p1, p2,..., p2019 be distinct primes. Prove that any group of
order p1p2...p2019 can be generated by 2019 elements. (Hint: Use
Lagrange's theorem.)
3. (10%) Show that lim φ(n)=∞ , where φ is the Euler's phi function.
n->∞
4. (20%) Let G be a group and H, K ne its finite subgroups. We define a
relation ~ on the Cartesian product H×K by the rule:(h,k)~(h',k')
if and only if hk=h'k' in G.
(i) Show that the relation ~ is an equivalence relation.
(ii) Given (h,k)εH×K, determine [(h,k)], the equivalence class containing
(h,k), and the order of [(h,k)].
|H||K|
(iii) Show that |HK|=