Let \( G \) be a group and let \( H \) and \( K \) be normal subgroups such that \( H \cap K = \{ e \} \). Let \[ \phi : G \to G/H \times G/K \] be the map \(\phi(g) = (Hg, Kg)\). Prove that the kernel of \(\phi\) is \(\{ e \}\).
Let \( G \) be a group and let \( H \) and \( K \) be normal subgroups such that \( H \cap K = \{ e \} \). Let \[ \phi : G \to G/H \times G/K \] be the map \(\phi(g) = (Hg, Kg)\). Prove that the kernel of \(\phi\) is \(\{ e \}\).
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 \( G \) be a group and let \( H \) and \( K \) be normal subgroups such that \( H \cap K = \{ e \} \). Let
\[
\phi : G \to G/H \times G/K
\]
be the map \(\phi(g) = (Hg, Kg)\).
Prove that the kernel of \(\phi\) is \(\{ e \}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2844e9b7-fdae-4ffb-b3b7-8f54bdb6d500%2F4bb67705-aaba-48e8-9688-a650b12d9292%2F9u95q4p_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( G \) be a group and let \( H \) and \( K \) be normal subgroups such that \( H \cap K = \{ e \} \). Let
\[
\phi : G \to G/H \times G/K
\]
be the map \(\phi(g) = (Hg, Kg)\).
Prove that the kernel of \(\phi\) is \(\{ e \}\).
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