- #1

AllRelative

- 42

- 2

## Homework Statement

let f

_{x}(z) = xz

let θ : Q

^{*}→ Aut(Q)

with θ(x) = f

_{x}

Is θ an isomorphism?

## Homework Equations

Homomorphisms,

Surjectivity and injectivity

## The Attempt at a Solution

θ is a homomorphism because

θ(xy) = f

_{xy}= f

_{x}* f

_{y}= θ(x)θ(y)

θ is injective because the neutral element of Aut(Q) is f

_{1}and

θ

^{-1}(f

_{1}) = 1 the neutral element of Q

^{*}

Prooving the surjectivity is where I am uncertain...

∀f

_{x}∈ Aut(Q) , There is a unique x ∈ Q

^{*}for which

θ(x) = f

_{x}

Is the codomain = Aut(Q) ?