I Showing that two groups are not isomorphic question

Mr Davis 97

I am trying to show that $\mathbb{R} - \{ 0\}$ is not isomorphic to $\mathbb{C} - \{0 \}$. If we simply look at $x^3 = 1$, it's clear that $\mathbb{R} - \{ 0\}$ has one solution while $\mathbb{C} - \{0 \}$ has three.

My question, how can I use $x^2 = -1$ to show that they are not isomorphic? Using $x^3 = 1$ is more clear because any isomorphism would preserve powers and preserve the identity. But using $x^2 = -1$ is less clear to me.

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fresh_42

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2018 Award
Interesting question. I would consider all elements of finite order. There are only two of them in $\mathbb{R}-\{0\}$ and infinitely many in $\mathbb{C}-\{0\}$. But if you only want to use the single relation $\varphi(r)=i$ it's a bit tricky, because one easily falls into unproven statements like the ordering of the two sets and an assumed isomorphism isn't necessarily order preserving. Do you have any ideas?

Edit: I think I got it: Consider $\varphi(-1)\cdot \varphi (-1)$.

"Showing that two groups are not isomorphic question"

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