# Counterexample to homomorphism problem

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• Jan 8th 2014, 08:44 AM
topsquark
Counterexample to homomorphism problem
Here it is:
Quote:

Show that if H is any group and h is an element of H with $h^n = e$, then there is a unique homomorphism from $Z_n = < x >$ to H such that $x \mapsto h$.
I have been able to prove such a homomorphism exists with a cyclic subgroup of H anyway. How do I extend this to they whole group H? For example when I do this using $H = D_8$ I end up with a "homomorphism" that is a relation, not a function.

And as far as uniqueness goes the homomorphism maps x to h. Isn't that the only function that will do that? I mean if we have $\phi : Z_n \to H: x \mapsto h$ how could we have a distinct function $\psi : Z_n \to H: x \mapsto h$? If they both map x to h wouldn't they have to be the same function automatically? I'm missing something here...

-Dan
• Jan 8th 2014, 09:34 AM
Deveno
Re: Counterexample to homomorphism problem
There is no requirement that the homomorphism be surjective....

The whole point of the problem is that: o(h)|n. If this were NOT so, we would have hn ≠ e, in which case:

φ(e) = φ(xn) = φ(x)n = hn ≠ e, so φ is not a homomorphism (which must map identity to identity).

Homomorphisms are just certain functions between groups, the image of a homomorphism does NOT have to be the entire target group.