# Properties of Homomorphism

1. Dec 14, 2011

### sleventh

I am wondering what are the possible homomorphisms
$\tau$ : $Z$$\overline{+}$ -> $Z$$\overline{+}$

From this it should be possible to determine which is injective, surjective, and which are isomorphic.

Homomorphisms between Z plus to Z plus will all be of the form $\tau$(x) = nx
since $\tau$(x) = $\tau$(1)$\underline{1}$ + $\tau$(1)$\underline{2}$ + ... + $\tau$(1)$\underline{x}$

since we have a homomorphism and x is one summed x times.

all are injective

now im not sure how to tell which are surjective

2. Dec 28, 2011

### DonAntonio

What kind of algebraic structure and under what operation(s) you think "Z plus" (the natural numbers, I presume?) is for you to talk about "homomorphisms"? Perhaps a monoid?
Tonio

3. Dec 28, 2011

### spamiam

I'm guessing he means the group $(\mathbb{Z},+)$, the group of integers under addition.

sleventh, you're right that each homomorphism can be written $\tau_n(x) = nx$ for an integer n. (Since $\mathbb{Z}$ is generated by 1, everything is determined by $\tau_n(1)$.) Can you write out the range of the homomorphism for each n? There shouldn't be very many that are surjective.

Also, not all are injective. I can think of one homomorphism (a boring one, admittedly) that isn't.