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## Homework Statement

Prove that Q+, the group of positive rational numbers under multiplication, is isomorphic to a proper subgroup of itself.

## Homework Equations

None

## The Attempt at a Solution

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Not at all sure if this is legit.

Let phi: Q

_{+}--> G

phi(x) = x

^{2}, x is in Q

_{+}

We will demonstrate that G c Q+

It is a subgroup: 1=e is in G, and ab

^{-1}= x

^{2}y

^{-2}= (xy

^{-1})

^{2}is in G

It is a proper subgroup: 2 is in Q+, but sqrt(2) is not in G and indeed not in Q+

One-to-one:

phi(x) = phi(y)

x

^{2}= y

^{2}

x, y > 0

x = y

Onto:

Take some g in G. We have that sqrt(g) satisfies phi(sqrt(g)) = sqrt(g)

^{2}= g.

Therefore, there is an element in Q+ such that phi(x)=g.

Operation preservation:

We have phi(x*y) = (xy)

^{2}= x^2y

^{2}

phi(x)phi(y) = x

^{2}y

^{2}

So phi(x*y)=phi(x)*phi(y)

Therefore, phi is an isomorphism between Q+ and a proper subgroup of itself.