AEAll(iii)(ii)NoneWhich of the following are true?(i): H = {e, (1 2)} is a normal subgroup of S3.(ii): 6Z is a normal subgroup of 3Z.(iii): H = {e, (1 2) (3 4)} is a normal subgroup of An for any n ≥ 4.

Answered: Image /qna-images/answer/ea46f479-4405-47a0-93d7-45f761857a16.jpg

Answered: A B D E All (iii) (i) (ii) None Which…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.5: Normal Subgroups

Problem 9E: 9. Consider the octic group of Example 3. Find a subgroup of that has order and is a normal...

See similar textbooks

Similar questions

Suppose G1 and G2 are groups with normal subgroups H1 and H2, respectively, and with G1/H1 isomorphic to G2/H2. Determine the possible orders of H1 and H2 under the following conditions. a. G1=24 and G2=18 b. G1=32 and G2=40
Show that An has index 2 in Sn, and thereby conclude that An is always a normal subgroup of Sn.
22. If and are both normal subgroups of , prove that is a normal subgroup of .
9. Consider the octic group of Example 3. Find a subgroup of that has order and is a normal subgroup of . Find a subgroup of that has order and is not a normal subgroup of .
2. Show that is a normal subgroup of the multiplicative group of invertible matrices in .
(2) Consider S4 and its subgroups H = {i,(12)(34),(13)(24),(14)(23)} and K ={i,(123),(132)}. For a = (1234) ∈ S4 find (H, K) and (K, H) double cosets
(6) Let G = S4 and σ = = (1, 2, 4)(1, 3). Write o as a product of disjoint cycles. Find the order of o. (b) Write o46 as a product of disjoint cycles. (c) Let H=. Is H a normal subgroup of G.
(1) Find all subgroupsof (Zs.+3). (2) Let (Zg.ts) be agroup and H-. Is H a subgroup of Zg? (3) If H={0,6,12.18}, show that (H.+24) is a cyclic subgroupof (Z2.+21). Also list the elements of each coset of H in Z24.
S What is [Z12: (4)]? Find all cosets of the subgroup (4) of Z12. A. {(4), 1 + (4)} B. {(4), 1+ (4), 2 + (4)} C. {(4), 1+ (4), 2 + (4), 3+ (4)} D. {(4), 1+ (4), 2 + (4), 3+ (4), 4+ (4)} Leto = (1 254) (23) E S5. Find the index of (o) in S5. 6 2
consider S3 and N = { (1), (12)} and H = { (1) , (123), (132)} both are subgroups of S3. Determine which one is normal
C H B D E Find the order of the subgroup generated by a for each of the following groups: (124) (i): S4, a (ii) (Z16, +), a = 6 (iii): U10,a = 7 (iv): (Z₁ x Z6, +), a =(2, 2) (i): 4; (ii): 16; (iii): 4; (iv): 12 (i): 3; (ii): 8; (iii): 3; (iv): 6 (i): 4; (ii): 8; (iii): 3; (iv): 24 (i): 3; (ii): 16; (iii): 5; (iv): 4 (i): 3; (ii): 8: (iii): 4; (iv): 6 :
D D (B) U P E (i), (ii) All (i), (iii) (ii), (iii) All Consider the subgroup 10Z of 2Z, which of the following are correct? (i): The cosets of 10Z in 2Z are 10Z, 2 + 10Z, 4+ 10Z,6 + 10Z,8 + 10Z. (ii): [2Z: 10Z] 5. (iii): 10Z is a normal subgroup of 2Z. :

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS

A B D E All (iii) (i) (ii) None Which of the following are true? (i): H = {e, (1 2)} is a normal subgroup of S3. (ii): 6Z is a normal subgroup of 3Z. (iii): H= {e, (1 2)(3 4)} is a normal subgroup of An for any n 24.