[수학]Exercises

1. 

p:odd prm

Show that p ≡ 1 (mod 4) iff x^2 ≡ (-1) (mod p) has an integer solution

(link)

2. 

G:finite group

H:subgroup of G

G=union of xHx^(-1) over all x in G

이면 G=H이다.

(link)

3. 

G:group

H:subgroup of G s.t. [G:H]<inf

이면 te N s.t. N:proper normal in G, [G:N]<inf

(link) 

4.

Prove that there is no simple group of order 525=3*5^2*7

(link)

5. 

|G|=21, non-cyclic group

Find the number of sylow 3-subgroups of G

(link)

6.

|G|=45=3^2*5, group

Prove that there is an element g in G s.t. |g|=15=3*5

(link)

7.

Find the number of sylow 7-subgroups of S_7

(link)

8. 

(TS,top1), (TS,top2), top1<top2일 때

(1) top1이 T2이면 top2도 T2인가?

(2) top1이 T3이면 top2도 T3인가?

(3) top1이 T4이면 top2도 T4인가?

(link)

9.

|G|=2^3 3^4 5^2 7인 finite abelian group일 때, elementary divisor로 classification, invariant factor로 classification(link)