Group Automorphism/Examples/Constant Product on Real Numbers

Examples of Group Automorphisms
Let $\struct {\R, +}$ denote the real numbers under addition.

Let $\alpha \in \R$ be a real number.

Let $f: \R \to \R$ be the real function defined as:
 * $\forall x \in \R: \map f x = \alpha x$

Then $f$ is a (group) automorphism $\alpha \ne 0$.

Proof
Let $\alpha \in \R$.

We have that:

Thus $f$ is a (group) homomorphism for all $\alpha \in \R$.

Sufficient Condition
Let $\alpha \in \R$ such that $\alpha \ne 0$.

Then $\dfrac 1 \alpha$ is the multiplicative inverse of $\alpha$ in the field of real numbers $\R$.

We have that:

and so $f$ is injective by definition.

Then we have:

This demonstrates that $f$ is surjective.

So by definition $f$ is a bijection.

Thus $f$ is an (group) isomorphism from $\R$ to $\R$ and so a (group) automorphism.

Necessary Condition
Let $f$ be a (group) automorphism.

$\alpha = 0$.

Then we have:


 * $\forall x, y \in \R: \map f x = \map f y = 0$

and so $f$ is not injective.

Hence $f$ is not a bijection.

Therefore $f$ is not an automorphism.

Hence by Proof by Contradiction $f$ is not an automorphism.