4. (103) Let G be a group with identity element e, and let H and K be subgroups of G.Assume that(i) H and K are normal in G,(ii) HNK = {e},(iii) G = {hk | h E H and k e K}.Prove that G is isomorphic to the direct product H × K.(Hint : First, prove that hkh-'k-1= e for all h e H and k e K.)

Answered: Image /qna-images/answer/ceebdd7f-743f-4f7a-b2a8-63145b3db74b.jpg

1. (103) Let G be a group with identity element e, and let H and K be subgroups of G. Assume that (i) H and K are normal in G, (ii) HNK = {e}, (iii) G= {hk | h € H and k € K}. Prove that G is isomorphic to the direct product H x K. (Hint : First, prove that hkh-'k-1 = e for all h e H and k E K.)

1. (103) Let G be a group with identity element e, and let H and K be subgroups of G. Assume that (i) H and K are normal in G, (ii) HNK = {e}, (iii) G= {hk | h € H and k € K}. Prove that G is isomorphic to the direct product H x K. (Hint : First, prove that hkh-'k-1 = e for all h e H and k E K.)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.4: Cosets Of A Subgroup

Problem 14E: Let H be a subgroup of a group G. Prove that gHg1 is a subgroup of G for any gG.We say that gHg1 is...

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