Let $ G $ and $ G' $ be groups.1. Show that if $ G $ and $ G' $ are abelian, then $ G \oplus G' $ is abelian.2. Show that if $ H \unlhd G $ and $ H' \unlhd G' $, then $ H \oplus H' \unlhd G \oplus G' $.3. Give a simple example to show that even if $ G $ and $ G' $ are cyclic, $ G \oplus G' $ may not be cyclic.4. Bonus: If $ H \unlhd G $ and $ H' \unlhd G' $, show that $ (G \oplus G')/(H \oplus H') \cong (G/H) \oplus (G'/H') $.

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G and G' are abelian, then G ® G' is abelian.

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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Question

Let $ G $ and $ G' $ be groups.

1. Show that if $ G $ and $ G' $ are abelian, then $ G \oplus G' $ is abelian.

2. Show that if $ H \unlhd G $ and $ H' \unlhd G' $, then $ H \oplus H' \unlhd G \oplus G' $.

3. Give a simple example to show that even if $ G $ and $ G' $ are cyclic, $ G \oplus G' $ may not be cyclic.

4. Bonus: If $ H \unlhd G $ and $ H' \unlhd G' $, show that $ (G \oplus G')/(H \oplus H') \cong (G/H) \oplus (G'/H') $.

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