# Automorphisms in the integer additive group

Consider

$$f_a:(\mathbb{Z},+) \rightarrow(\mathbb{Z},+), f_a(k)=ka, \forall k \in\mathbb{Z}$$

the endomorphisms in the integer additive group.

I have to prove that there are only two automorphisms in this group, $$f_{-1}, f_1$$. While the injectivity is obvious, I can't figure out the surjectivity part.

• I think of an automorphism as a homomorphism that has an inverse which is also a homomorphism. What could an inverse of $f_a$ possibly be? – Lord Shark the Unknown Mar 30 at 11:09

Note that any homomorphism $$f:(\mathbb{Z},+)\to(\mathbb{Z},+)$$ is determined by $$f(1)$$, since $$f(n)=f(1)+\cdots +f(1)=nf(1)$$ . Then, if $$f(1)=a$$, the image of $$f$$ is going to be the integer multiples of $$a$$, which is only $$\mathbb{Z}$$ when $$a=\pm 1$$.
Let $$f_a(x) = ax$$. Notice that the image of this function will always generate multiples of $$a$$. That is, $$f_a(\mathbb{Z}) = a\mathbb{Z}$$. Note that $$a\mathbb{Z} = \mathbb Z \iff a = \pm 1$$.