- #1

- 42

- 2

## Homework Statement

*This is from a Group Theory class

**My secondary aim is to practice writing the math perfectly because I tend to loose a lot of points for not doing so in exams...

Let λ ∈ Q

^{*}

f

_{λ}: Q → Q defined as f

_{λ}(x) = λx

a) Show that f

_{λ}is and automorphism of the group of rationals with the sum operation (Q,+)

## Homework Equations

Let f: (G,*) → (G', ⋅)

e is the neutral element of G

e' is the neutral element of G'

The definition of homomorphisms:

f is a homomorphism if

f(x*y) = f(x) ⋅ f(y) , ∀x,y ∈ G

f is injective ⇔ ker(f) = {e}

f is surjective ⇔ Im(f) = G'

## The Attempt at a Solution

Let's show that f

_{λ}is a homomorphism.

Let x,y ∈ Q

f

_{λ}(x+y) = λ(x+y) = λx + λy = f

_{λ}(x) + f

_{λ}(y)

which proves that f

_{λ}is a homomorphism.

Let's show that f

_{λ}is injective.

The neutral element of Q is 0.

0 = λx ⇒ x = 0 because λ ≠ 0

We can therefore write Ker(f

_{λ}) = {0} which proves that f

_{λ}is injective.

Let's show that f

_{λ}is surjective.

Every element y of Q can be written with the form λx where x∈Q.

This means that Im(f

_{λ}) = Q.

This proves that f

_{λ}is surjective.

Finally, we proved that f

_{λ}is a endomorphism that is injective and surjective. This is the definition of an automorphism.