What's the number of group isomorphisms from the group (Z, +) to itself??
my approach: I think it has to be infinity, but that is an incorrect answer...
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1Think about $f(1)$ . Since $f$ is an isomorphism (proper term is automorphism) then where should $f(1)$ go ? – user-492177 Oct 06 '20 at 04:09
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Use the group automorphism axioms / definition and you should see that it will need to fix $0$ as the additive identity. This answer depends on the precise type of isomorphism and whether you need to fix $0$ as the identity or whether in your morphed group you could have e.g. $1$ as the additive identity instead. – it's a hire car baby Oct 06 '20 at 13:32
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Any homomorphism of a cyclic group is determined by what it does to a generator. So, by what it does to $1$.
But to be an automorphism, it has to be surjective. This won't happen unless $1$ is mapped to a generator.
So we need $\varphi(1)=1$, or $\varphi(1)=-1$. Thus there are two.
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This depends upon the theorem that ${-1,1}$ are the only generators of $(\Bbb Z,+)$ which the OP may not already have. – it's a hire car baby Oct 06 '20 at 13:35
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1@samerivertwice This means we have to prove that $1/n$ isn't an integer for $n\ne\pm1$. Point taken. – Oct 06 '20 at 20:23
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OP would appear to think that $x\mapsto x+i$ is an automorphism for any $i\in\Bbb Z$. – it's a hire car baby Oct 06 '20 at 21:01
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Yes, or some such. But of course the only one of those that works is for $i=0$. @samerivertwice – Oct 06 '20 at 21:05