Exercise 3 (Second isomorphism theorem). Let G be a group, N < G, H < G. Prove that(a) Hn N < H.(b) N< ΝΗ.(c) H/(HN N)~ (NH)/N.(Hint: Construct a homomorphism H→(NH)/N. Then apply the first isomorphism theo-rem.)

Let, G be group N∆G, H<G Part(a): H∩N∆H⇒e∈H and e∈H∩N⇒e∈H∩N Let, g∈H∩N then for every x∈H…

Exercise 3 (Second isomorphism theorem). Let G be a group, N < G, H < G. Prove that (а) НnN4Н. ( b) N ΝΗ. (c) Η/ (ΗnN) < (NH)/Ν. (Hint: Construct a homomorphism H→(NH)/N. Then apply the first isomorphism theo- rem.)

Exercise 3 (Second isomorphism theorem). Let G be a group, N < G, H < G. Prove that (а) НnN4Н. ( b) N ΝΗ. (c) Η/ (ΗnN) < (NH)/Ν. (Hint: Construct a homomorphism H→(NH)/N. Then apply the first isomorphism theo- rem.)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 24E

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Exercise 3 (Second isomorphism theorem). Let G be a group, N < G, H < G. Prove that
(a) Hn N < H.
(b) N< ΝΗ.
(c) H/(HN N)~ (NH)/N.
(Hint: Construct a homomorphism H→(NH)/N. Then apply the first isomorphism theo-
rem.)

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