(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find all homomorphisms [itex]f: \mathbb{Z},+ \rightarrow \mathbb{Z},+[/itex]. Determine which are injective, which are surjective, and which are isomorphisms.

Note.I must prove everything.

2. Relevant equations

Notation.[itex]\mathbb{Z}n = \{ p : p = kn, \, \, \, \mathrm{k} \in \mathbb{Z} \}[/itex]

3. The attempt at a solution

I'm not sure if I have found all homomorphisms from [itex]\mathbb{Z},+[/itex] to itself, but thus far I found a family of homomorphisms from [itex]\mathbb{Z},+[/itex] to itself.

Define [itex]f: \mathbb{Z},+ \rightarrow \mathbb{Z},+[/itex] such that [itex]f(x) = nx[/itex], where [itex]n \in \mathbb{Z}[/itex]. This is the homomorphism from [itex]\mathbb{Z},+[/itex] to it's subgroup [itex]\mathbb{Z}n[/itex].

Proof that f is a homomorphism.Let both [itex]x, y \in \mathbb{Z}[/itex]. Then [itex]f(x+y) = n(x+y) = nx + ny = f(x) + f(y)[/itex]. Done.

I've determined that every homomorphism of form [itex]f[/itex], as defined above, is injective. I'll claim that [itex]\mathrm{ker}(f) = \{ 0 \}[/itex]. To check this, I'll suppose that there exists another element, say [itex]k[/itex], such that [itex]f(k) = 0[/itex]. Using the definition of [itex]f[/itex], it follows that [itex]nk = 0[/itex] [itex]\Rightarrow[/itex] [itex] k = 0[/itex]. Thus [itex]\mathrm{ker}(f) = \{ 0 \}[/itex].

I've also determined that the only [itex]f[/itex] that is surjective is [itex]f(x) = x[/itex]. To prove this, I'll suppose that there exists another such [itex]f[/itex] that is surjective, say [itex]f(x) = kx[/itex], with [itex]k \neq 1[/itex]. Note that [itex]k \in \mathbb{N}[/itex]. Set [itex]f(x) = k+1[/itex] [itex]\Rightarrow[/itex] [itex]kx = k+1[/itex] [itex]\Rightarrow[/itex] [itex]x = 1 + \frac{1}{k}[/itex] which is never an integer if [itex]k \neq 1[/itex], a contradiction since [itex]x \in \mathbb{Z}[/itex].

Thus from the information above, it follows that the only isomorphism is the homomorphism [itex]f[/itex] as defined above, such that [itex]f(x) = x[/itex].

My question is have I done my work correctly and are there any other homomorphisms from [itex]\mathbb{Z},+[/itex] to itself?

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# Homework Help: Describe all homomorphisms

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