Suppose that the group (H,#) is a homomorphic image of the group (G,*). Is (G,*) necessarily a homomorphic image of the group (H,#)?

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- Oct 28th 2009, 05:34 PMjujabHomomorphic Image Question
Suppose that the group (H,#) is a homomorphic image of the group (G,*). Is (G,*) necessarily a homomorphic image of the group (H,#)?

- Oct 28th 2009, 07:21 PMBruno J.
No! $\displaystyle H$ could be smaller than $\displaystyle G$, and then there is no way $\displaystyle G$ could be the image of any kind of function defined on $\displaystyle H$.