Group Theory

Multiplicative Groups of Real Numbers and Complex Numbers are not Isomorphic

Group Theory Problems and Solutions in Mathematics

Problem 130

Let $\R^{\times}=\R\setminus \{0\}$ be the multiplicative group of real numbers.
Let $\C^{\times}=\C\setminus \{0\}$ be the multiplicative group of complex numbers.
Then show that $\R^{\times}$ and $\C^{\times}$ are not isomorphic as groups.

Add to solve later

Recall.

Let $G$ and $K$ be groups.
Recall that a map $f: G \to K$ is a group homomorphism if
\[f(ab)=f(a)f(b)\] for all $a, b \in G$.
A group isomorphism is a bijective homomorphism.
If there is a group isomorphism from $G$ to $K$, we say that $G$ and $K$ are isomorphic (as groups).

We give two proofs.

Proof 1.

Seeking a contradiction, assume that there is a group isomorphism $\phi: \C^{\times} \to \R^{\times}$.
Since $\phi$ is a group homomorphism, $\phi(1)=1$. Thus we have
\[1=\phi(1)=\phi((-1)(-1))=\phi(-1)\phi(-1)=\phi(-1)^2.\] Hence $\phi(-1)=\pm 1$. But since $\phi$ is injective and $\phi(1)=1$, we must have $\phi(-1)=-1$.

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

Since $\phi(i)\in \R^{\times}$, $\phi(i)^2$ must be a positive number. Thus we reached a contradiction.
Hence there is no isomorphism between $\R^{\times}$ and $\C^{\times}$.

Proof 2.

Suppose that there is a group isomorphism $\phi: \C^{\times} \to \R^{\times}$.
We want to find a contradiction.
Let $\zeta=e^{2\pi i /3}$ be a primitive third root of unity.

Since $\zeta^3=1$ and $\phi$ is a group homomorphism, we have
\begin{align*}
1=\phi(1)=\phi(\zeta^3)=\phi(\zeta)^3.
\end{align*}
Since $\phi(\zeta)$ is a real number, this implies that $\phi(\zeta)=1$.

This is a contradiction since $\phi$ is injective, but we have $\phi(1)=1=\phi(\zeta)$.
Therefore, there cannot be a group isomorphism between $\R^{\times}$ and $\C^{\times}$.

Add to solve later

More from my site

$The Additive Group $\R$ is Isomorphic to the Multiplicative Group $\R^{+}$ by Exponent Function$ The Additive Group $\R$ is Isomorphic to the Multiplicative Group $\R^{+}$ by Exponent Function Let $\R=(\R, +)$ be the additive group of real numbers and let $\R^{\times}=(\R\setminus\{0\}, \cdot)$ be the multiplicative group of real numbers. (a) Prove that the map $\exp:\R \to \R^{\times}$ defined by \[\exp(x)=e^x\] is an injective group homomorphism. (b) Prove that […]
The Additive Group of Rational Numbers and The Multiplicative Group of Positive Rational Numbers are Not Isomorphic Let $(\Q, +)$ be the additive group of rational numbers and let $(\Q_{ > 0}, \times)$ be the multiplicative group of positive rational numbers. Prove that $(\Q, +)$ and $(\Q_{ > 0}, \times)$ are not isomorphic as groups. Proof. Suppose, towards a […]
The Center of the Heisenberg Group Over a Field $F$ is Isomorphic to the Additive Group $F$ Let $F$ be a field and let \[H(F)=\left\{\, \begin{bmatrix} 1 & a & b \\ 0 &1 &c \\ 0 & 0 & 1 \end{bmatrix} \quad \middle| \quad \text{ for any} a,b,c\in F\, \right\}\] be the Heisenberg group over $F$. (The group operation of the Heisenberg group is matrix […]
Group of $p$-Power Roots of 1 is Isomorphic to a Proper Quotient of Itself Let $p$ be a prime number. Let \[G=\{z\in \C \mid z^{p^n}=1\} \] be the group of $p$-power roots of $1$ in $\C$. Show that the map $\Psi:G\to G$ mapping $z$ to $z^p$ is a surjective homomorphism. Also deduce from this that $G$ is isomorphic to a proper quotient of $G$ […]
$The Set of Square Elements in the Multiplicative Group $(\Zmod{p})^*$$ The Set of Square Elements in the Multiplicative Group $(\Zmod{p})^*$ Suppose that $p$ is a prime number greater than $3$. Consider the multiplicative group $G=(\Zmod{p})^*$ of order $p-1$. (a) Prove that the set of squares $S=\{x^2\mid x\in G\}$ is a subgroup of the multiplicative group $G$. (b) Determine the index $[G : S]$. (c) Assume […]
Normal Subgroups, Isomorphic Quotients, But Not Isomorphic Let $G$ be a group. Suppose that $H_1, H_2, N_1, N_2$ are all normal subgroup of $G$, $H_1 \lhd N_2$, and $H_2 \lhd N_2$. Suppose also that $N_1/H_1$ is isomorphic to $N_2/H_2$. Then prove or disprove that $N_1$ is isomorphic to $N_2$. Proof. We give a […]
Inverse Map of a Bijective Homomorphism is a Group Homomorphism Let $G$ and $H$ be groups and let $\phi: G \to H$ be a group homomorphism. Suppose that $f:G\to H$ is bijective. Then there exists a map $\psi:H\to G$ such that \[\psi \circ \phi=\id_G \text{ and } \phi \circ \psi=\id_H.\] Then prove that $\psi:H \to G$ is also a group […]
$Group Homomorphism from $\Z/n\Z$ to $\Z/m\Z$ When $m$ Divides $n$$ Group Homomorphism from $\Z/n\Z$ to $\Z/m\Z$ When $m$ Divides $n$ Let $m$ and $n$ be positive integers such that $m \mid n$. (a) Prove that the map $\phi:\Zmod{n} \to \Zmod{m}$ sending $a+n\Z$ to $a+m\Z$ for any $a\in \Z$ is well-defined. (b) Prove that $\phi$ is a group homomorphism. (c) Prove that $\phi$ is surjective. (d) Determine […]