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15. Let (G, *) be a group, and let H₁, H₂,..., Hk be normal subgroups of G such that H₁ H₂0... Hk = {e}. Prove that there is a monomorphism: G G/H₁ x G/H₂ x...G/H. 16. Let (G, *) be a group, and let H ≤ G, K≤G. Prove that H/(HK) H✩ K/K.

15. Let (G, *) be a group, and let H₁, H₂,..., Hk be normal subgroups of G such that H₁ H₂0... Hk = {e}. Prove that there is a monomorphism: G G/H₁ x G/H₂ x...G/H. 16. Let (G, *) be a group, and let H ≤ G, K≤G. Prove that H/(HK) H✩ K/K.

Advanced Engineering Mathematics
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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15. Let (G, *) be a group, and let H₁, H₂,..., Hk be normal subgroups of G such that H₁ H₂0... Hk = {e}. Prove that there is a monomorphism : G G/H₁ x G/H₂ x...G/H. 16. Let (G, *) be a group, and let H ≤ G, K≤G. Prove that H/(H^K) ≈ H⋆ K/K.
Transcribed Image Text:15. Let (G, *) be a group, and let H₁, H₂,..., Hk be normal subgroups of G such that H₁ H₂0... Hk = {e}. Prove that there is a monomorphism : G G/H₁ x G/H₂ x...G/H. 16. Let (G, *) be a group, and let H ≤ G, K≤G. Prove that H/(H^K) ≈ H⋆ K/K.
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