Given the group (G, *).(a) Show that for nonempty HCG, then (H, *) (G, *)a,b H⇒ a*b¹ € H.(b) For some fixed element a € G, define the set C(a) = {re G: a* r = r * a}. Prove that(C(a), *) (G, *).(c) Define the set Z(G) = {re G: a*x=x*a for every a E G}. Prove that (2, *) (G, *).[Here, the symbol means subgroup. You may suppress the operation and use productnotation, i.e, write ab-¹ to mean a * b-¹, etc.]

Answered: Image /qna-images/answer/b73707b8-2d8b-416f-bc12-16d928f5e33b.jpg

(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) = {rG: a* x = x *a}. Prove that (C(a), *) (G, *). (c) Define the set Z(G) = {z EG: a*x=x*a for every a E 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.]

(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) = {rG: a x = x a}. Prove that (C(a), ) (G, ). (c) Define the set Z(G) = {z EG: ax=xa for every a E 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

Chapter3: Groups

Section3.1: Definition Of A Group

Problem 42E: 42. For an arbitrary set , the power set was defined in Section by , and addition in was...

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