Homomorphism of Groups and their generators

**smallgun** · December 1st 2008, 12:53 PM

I know that f(g) is a generator of the image of f, say f(G).

Is it true that given a homomorphism f from a cyclic group G to a goup G' and a generator g of G, f(g) is a generator of G'?

I think it is positive, but need a concere proof.

Thank you for considering it.

**ThePerfectHacker** · December 1st 2008, 01:15 PM

Originally Posted by smallgun

Is it true that given a homomorphism f from a cyclic group G to a goup G' and a generator g of G, f(g) is a generator of G'?

Let $\phi : \mathbb{Z} \to \mathbb{Z}$ be $\phi (x) = 0$ where $\mathbb{Z}$ are the integers under addition.
Check that $\phi$ is a homomorphism.
Not $1$ generates $\mathbb{Z}$ but $\phi(1) = 0$ doth not generate $\mathbb{Z}$ .

However, if $f : G\to G'$ is an isomorphism then this is true.

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