Additive Group and Multiplicative Group of Field are not Isomorphic
Theorem
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$.
Let $\struct {F, +}$ denote the additive group of $F$.
Let $\struct {F_{\ne 0_F}, \times}$ denote the multiplicative group of $F$.
Then $\struct {F, +}$ and $\struct {F_{\ne 0_F}, \times}$ are not isomorphic to each other.
Proof
Aiming for a contradiction, suppose $\phi: \struct {F_{\ne 0_F}, \times} \to \struct {F, +}$ is an isomorphism.
By definition:
- $0_F$ is the identity of $\struct {F, +}$
and
- $1_F$ is the identity of $\struct {F_{\ne 0_F}, \times}$.
We have that:
\(\ds 0_F\) | \(=\) | \(\ds \map \phi {1_F}\) | Epimorphism Preserves Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi {\paren {-1_F} \times \paren {-1_F} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi {-1_F} + \map \phi {-1_F}\) | Definition of Group Isomorphism |
and so by definition $F$ has characteristic $2$.
Let $x \in \struct {F_{\ne 0_F}, \times}$.
Then:
\(\ds \map \phi {x^2}\) | \(=\) | \(\ds \map \phi x + \map \phi x\) | Definition of Group Isomorphism | |||||||||||
\(\ds \) | \(=\) | \(\ds 0_F\) | as $F$ has characteristic $2$ |
As $\phi$ is an isomorphism, it is also a monomorphism.
From Kernel is Trivial iff Monomorphism:
- $\map \ker \phi = \set {1_F}$
and so $x^2 = 1$.
Thus $x = 1$ and so $\order F = 2$.
Thus $\order {F_{\ne 0_F} } = 1$.
So $\phi$ is a bijection from a set of cardinality $1$ to a set of cardinality $2$.
So $\phi$ cannot be a bijection and so cannot be an isomorphism.
Hence the result by Proof by Contradiction.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $9$: Rings: Exercise $16$