Which Homomorphisms are Injective and Surjective between Z plus and Z plus?

[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 I am not sure how to tell which are surjective

 

[DonAntonio]

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 I am not sure how to tell which are surjective

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

 

[spamiam]

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

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.