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(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 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.]

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 product notation, i.e, write ab-¹ to mean a * b-¹, etc.]
Transcribed Image Text: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 product notation, i.e, write ab-¹ to mean a * b-¹, etc.]
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