# Algebra questions, (emergency)

1. How many homomorphism are there of $$\mathbb{Z}$$ onto $$\mathbb{Z}$$?

I don't know why this is the answer! Is it because the only homomorphism possible are $$\phi(x) = x$$ and $$\phi(x) = e$$ (where e is the identity) ?

Anything else won't work, for example: if
$$\phi(x)=2x$$

then $$\phi(xy) = 2xy$$

and $$\phi(x)\phi(y) = 2x2y$$

$$2xy \neq 2x2y.$$ ?

Or should I be using additive notation because $$\mathbb{Z}$$ is the set of integers? Okay next question:

2. Let $$\phi: G \rightarrow G'$$ be a group homomorphism. Show that if $$|G|$$ is finite, then $$|\phi[G]|$$ is finite and is a divisor of $$|G|$$.

Homomorphisms don't need to be one-to-one or onto, so I'm havinga hard time seeing why this must be true. Why can't each element in G map to an infinite number of items in G'? My book uses a diagram that suggests that this kind of mapping is like a projection to the x axis, so the set G may have two dimensions, but the set G' has only one. I don't see why this has to be the case, though ...

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