Let $\mathbb{C}$ denote the field of complex numbers.
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Is there an isomorphism $$\underset{\text{Countable}}{\mathbb{C}\times\mathbb{C}\times\ldots} \rightarrow \mathbb{C}$$
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Is there an isomorphism as above if each $\mathbb{C}$ is just a multiplicative monoid.
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Let $\mathbb{C}^\times=\mathbb{C}\backslash 0$, is there an isomorphism
$$\underset{\text{Countable}}{\mathbb{C}^\times\times\mathbb{C}^\times\times\ldots} \rightarrow \mathbb{C}^\times$$
I believe there should an isomorphism atleast in terms of cardinality. But, I have forgotten how to prove the above statements. This question is linked to Does $\varprojlim_{t\mapsto t^p}\mathbb{C}=\mathbb{C}$?
where I show that the left hand side is a multiplicative monoid of the form $$\underset{\text{Countable}}{\mathbb{C}\times\mathbb{C}\times\ldots} $$ and I seek an isomorphism to the right.
Best Answer
2)) No, because the monoid from the right-hand side has no zero divisors, whereas the monoid from the right-hand side has.
3)) Monoids (in fact, groups) from the left-hand side and the right-hand side are not isomorphic, because for each prime $p$ the former has $\frak c$ elements of order $p$ , whereas the latter has only $p$ such elements.