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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.

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  • $\begingroup$ 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? $\endgroup$ – Lord Shark the Unknown Mar 30 at 11:09
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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 $a=\pm 1$.

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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 $a\mathbb{Z} = \mathbb Z \iff a = \pm 1$.

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