7. Let (G,) be a group and let a G. DefinefloralC(a)= {bEG: ab = ba}Z(G) = {bEG: ab = ba for all a € G}Prove that(a). C(a) ≤ G,(b). Z(G) ≤ G,(c). Z(G) = C(a),aЄG(d). G abelian ⇒ Z(G) = G.Dr.[This is called the centralizer of a]This is called the center of G].Alabb

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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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7. Let (G, *) be a group and let a G. Define C(a)= {bEG: ab ba Z(G) = {bEG: ab-ba for all a € G} Prove that (a). C(a) ≤ G, (b). Z(G) ≤ G, (c). Z(G) = C(a), a€G aco [This is called the centralizer of al This is called the center of G]. Alabb