Consider the cyclic group G={a,a^2,a^3,...a^12=(adsbygoogle = window.adsbygoogle || []).push({}); u} and its subgroup G`={a^2,a^4,...,a^12}. My book says that the mapping

a^n ---> a^2n is an homomorphism of G onto G` (this seems true)

and that X: a^n ---> a^n is homomorphism of G` onto G (this seems to be false to me, a misprint)

A homomorphism of G` onto G would have

1)every g` in G` has a unique image g in G (true)

2)if X(a`)=a and X(b`)=b then X(a` o b`)=X(a x b) with operator o for G` and x for G (true)

3)every g in G is an image (false) e.g. a^3 in G is not the image of any g` in G`.

In fact, it would seem to me that you can have a homomorphism of a group onto a subgroup, but you could not have a homomorphism of a subgroup onto a group ever. Is this right, or am I missing something?

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# A question on homomorphisms

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