Prove that the multiplicative groups R-{0} and C-{0} are not isomorphic

The Mathematician — Sun, 05 Feb 2023 13:00:11 +0000

Prove that the multiplicative groups $\mathbb{R}-{0}$ and $\mathbb{C}-{0}$ are not isomorphic

Proof. Assume that

\mathbb{R}-{0}

is isomorphic to

\mathbb{C}-{0}

. Then there exists a mapping

\begin{align*} \phi: \mathbb{C}-{0} \longrightarrow \mathbb{R}-{0} \end{align*}

which is bijective and is a group homomorphism. By the definition of a group homomorphism, we have that

\phi(1) = 1

. We can also rewrite that as:

\begin{align*} 1 = \phi(1) = \phi((-1)(-1)) = \phi(-1)\phi(-1) = \phi(-1)^2. \end{align*}

Since

\phi

is injective, it must hold that

\phi(-1) = -1

. Again, we can rewrite that as:

\begin{align*} -1 = \phi(-1) = \phi(i^2) = \phi(i)^2. \end{align*}

So this means that

\phi(i)^2 \in \mathbb{R}^{\times}

. But that is not possible since

\phi(i)^2

must be positive number in

\mathbb{R}^{\times}

, and therefore

\phi(i)^2 \not \in \mathbb{R}^{\times}

. A contradiction. So, the multiplicative groups

\mathbb{R}-{0}

and

\mathbb{C}-{0}

are not isomorphic.

multiplicative groups R-{0} and C-{0} are not isomorphic Archives - Epsilonify