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

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

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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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 \( (G \oplus G')/(H \oplus H') \cong (G/H) \oplus (G'/H') \).
Transcribed Image Text: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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