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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.2011-05-31

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No, it does not. Take $H=G$, $g=id$, and $h$ the trivial homomorphism.