Given the group (G, *). 1 (a) Show that for nonempty HCG, then (H, +) (G,) a,b Ha*b¹ € H. (b) For some fixed element a € G, define the set C(a)= {G: ax=xa}. Prove that (C(a), *) (G, *). (c) Define the set Z(G) = {z EG: ax =a*a for every a G}. Prove that (Z, *) ≤ (G, *). [Here, the symbol means subgroup. You may suppress the operation and use product notation, i.e, write ab-¹ to mean a b-¹, etc.]
Given the group (G, *). 1 (a) Show that for nonempty HCG, then (H, +) (G,) a,b Ha*b¹ € H. (b) For some fixed element a € G, define the set C(a)= {G: ax=xa}. Prove that (C(a), *) (G, *). (c) Define the set Z(G) = {z EG: ax =a*a for every a G}. Prove that (Z, *) ≤ (G, *). [Here, the symbol means subgroup. You may suppress the operation and use product notation, i.e, write ab-¹ to mean a b-¹, etc.]
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.8: Some Results On Finite Abelian Groups (optional)
Problem 14E: Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic...
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![Given the group (G, *).
1
(a) Show that for nonempty HCG, then (H, *) (G, *) a,b Hab¹ € H.
(b) For some fixed element a € G, define the set C(a)= {G:ax-xa}. Prove that
(C(a), *) (G, *).
(c) Define the set Z(G)- (EG: az = a*a for every a € G}. Prove that (Z, *) (G, *).
[Here, the symbol means subgroup. You may suppress the operation and use product
notation, i.e, write ab-¹ to mean a b-¹, etc.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4cb4d097-3492-4cc6-a3d0-40439978c013%2F4532809e-03ac-4149-939d-6ab9b53884bb%2Fir152r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Given the group (G, *).
1
(a) Show that for nonempty HCG, then (H, *) (G, *) a,b Hab¹ € H.
(b) For some fixed element a € G, define the set C(a)= {G:ax-xa}. Prove that
(C(a), *) (G, *).
(c) Define the set Z(G)- (EG: az = a*a for every a € G}. Prove that (Z, *) (G, *).
[Here, the symbol means subgroup. You may suppress the operation and use product
notation, i.e, write ab-¹ to mean a b-¹, etc.]
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