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Let $\phi: G_{1} \rightarrow G_{2}$ be a homomorphism and $G_{1} = \langle a \rangle$ and $G_{2} = \langle b \rangle$ be infinite cyclic groups. Must we have $\phi(a) = b$?

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    OH I think the question should have been about isommorphisms not homomorphisms2012-12-04

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No every $\phi:G_1\rightarrow G_2$ with $\phi(\alpha)=(\beta)^k$ is a monomorphism of groups for every $k\in\mathbb{Z}$ ( $k\neq 0$, if $k=0$ then it's the trivial homomorpishm)

If you have $k=1$ or $k=-1$ then $\phi$ is an isomorphism of groups.

If what bothers you is “how we can fit” one infinite cyclic group into another infinite cyclic group if we don't map one generator to the other. Then just remember that every subgroup of an infinite cyclic group is also infinite cyclic.

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No, of course not. For example, let $G_1=G_2=\Bbb{Z}, a=b=1$. Then for any integer $k\neq1$, $\phi(n)=kn$ is a counterexample.

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As a counterexample, consider the zero homomorphism, that takes any generator to zero.

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Maybe you meant isomorphism. But even then, for $G_1=G_2=\Bbb{Z}/\Bbb{Z/p}, a=b=1$, $\phi(n)=kn$ with any $k$ coprime to $p$ and $k$ not congruent $1 \mod p$ is a counterexample.