3) Let G and H be two groups with e and e' as identities. Define P = G x H byP = G x H = { (g, h) | g € G,h E H }with algebraic operation (91, h4)(g2, h2) = (9192, h,h2). Prove that the set P' ={(g', h') I g' e Z(G),h' E Z(H)} is a subgroup of P. Also show that P' is the centerof P.(9)

Answered: Image /qna-images/answer/80c61d96-3f46-4827-b208-34779c2550f4.jpg

3) Let G and H be two groups with e and e' as identities. Define P = G x H by P = G x H = { (g,h) | g € G,h € H} with algebraic operation (g1, h;)(92, h2) = (9192, h,h2). Prove that the set P' = {(g', h') I g' E Z(G), h' € Z(H)} is a subgroup of P. Also show that P' is the center of P. (9)

3) Let G and H be two groups with e and e' as identities. Define P = G x H by P = G x H = { (g,h) | g € G,h € H} with algebraic operation (g1, h;)(92, h2) = (9192, h,h2). Prove that the set P' = {(g', h') I g' E Z(G), h' € Z(H)} is a subgroup of P. Also show that P' is the center of P. (9)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.6: Quotient Groups

Problem 23E

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3) Let G and H be two groups with e and e' as identities. Define P = G x H by
P = G x H = { (g, h) | g € G,h E H }
with algebraic operation (91, h4)(g2, h2) = (9192, h,h2). Prove that the set P' =
{(g', h') I g' e Z(G),h' E Z(H)} is a subgroup of P. Also show that P' is the center
of P.
(9)

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