Let G be an (additive) abelian group with subgroups H and K Show that G H K if and only if there are homomorphisms such that 1 L 1 1 H , 2 L 2 1 K , 1 L 2 0 and 2 L 1 0, where 0 is the map sending every element onto the zero (identity) element, and L 1 1 (x) L 2 2 (x) x for all x G H 41 G K 42...

Suppose that G H K Then there exist homomorphisms H K G and G H K such that 1HK and 1G Let 1 H K H a ...

Let G be an (additive) abelian group with subgroups H and K. Show that G H

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Let G be an (additive) abelian group with subgroups H and K. Show that G ≅ H ⊕ K if and only if there are homomorphisms

such that π₁L₁ = 1_H, π₂L₂ = 1_K, π₁L₂ = 0 and π₂L₁ = 0, where 0 is the map sending every element onto the zero (identity) element, and L₁π₁(x) + L₂π₂(x) = x for all x ϵ G.

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Algebra Graduate Texts In Mathematics 73

ISBN: 9780387905181

8th Edition

Authors: Thomas W. Hungerford

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Question Posted: Apr 13, 2023 03:08 AM

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Let G be an (additive) abelian group with subgroups H and K. Show that G H

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H ܙ 41 G K 42

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Suppose that G H K Then there exist homomorphisms H K G and G H K such that 1HK and 1G Let 1 H K H a...View the full answer

Algebra Graduate Texts In Mathematics 73

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