Question

Let De be the dihedral group of order 12. In other words, Do = {e,r,r2, m3, 74, 75, s, rs, rs, rºs, r*s, r® s} where p6 = 92 = e and sr = r-1s. a. Is H = {1, s, sr, sr2} a subgroup of Do? Why or why not? b. Is K = {1, 8, 73, r3s} a subgroup of Do? Why or why not?

Answer 1

solution De= şe,r, 82, 83, 84, 85, 8,788, 822, 23, 24, 2532 where ro= 8²=e & sr=rts. @ H = {1,.3, so, 3823 is not a subgroup of Do because not closed under the operation of De as si srth But so sr=8r=r&H. Bu that H is not satisfying closure Property Junce His not a subgroup of Do.
1
6. K={1,8, 83, r3s} is a subgroup of Do. because K is finite and also k is closed under Иe Swiskөч 9 ) , С ісц -I- - Bir?.-735 EK 4 7032 (03)-1 C as go=e. (13.693 (0348 = x38] Аз'я: 13 eк Г. К. та] 83 (833) = rosek [?r=e] Aldo (33 8 03) = 73 (73) 18 sek. Thus K is closed under the operation of Ds so that by finite subgroup Test ik is a subgroup of Da Finite subgroup Test: Any finite subset of Group G, which - is closed under the operation of į is subgroup of G. C 8?re.
2