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let $G$ be an abelian group. and $f:G\rightarrow \{0\}$ be the trivial homomorphism. suppose there exists $G\stackrel{g}{\rightarrow} H \stackrel{h}{\rightarrow} \{0\}$ such that $f=h\circ g$ does this imply that necessarely $H=\{0\}$ if not then under what condition this is true?

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I noticed that you've asked 7 questions, all of which have been answered, but you haven't accepted any answers, nor have you even clicked on an "up arrow" to vote for an answer. People here are putting time in to addressing your questions; you should at least vote for answers that are helpful, and accept at least some of the answers offered. – amWhy May 31 '11 at 1:56
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No, it does not. Take $H=G$, $g=id$, and $h$ the trivial homomorphism.

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