Prove (Q+, *) is isomorphic to a proper subgroup of itself

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In summary: I have forwarded your suggestion to my professor, and I am sure it will be helpful for the students too. Keep up the good work!In summary, we use the function phi(x) = x^2 to prove that Q+, the group of positive rational numbers under multiplication, is isomorphic to a proper subgroup of itself. The proof involves showing that the function is one-to-one, onto, and preserves the operation. However, there are some minor issues with the notation and definitions that need to be addressed.
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gummz
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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


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The Attempt at a Solution


[/B]
Not at all sure if this is legit.

Let phi: Q+ --> G
phi(x) = x2, x is in Q+
We will demonstrate that G c Q+
It is a subgroup: 1=e is in G, and ab-1 = x2 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)
x2 = y2
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^2y2
phi(x)phi(y) = x2y2
So phi(x*y)=phi(x)*phi(y)

Therefore, phi is an isomorphism between Q+ and a proper subgroup of itself.
 
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Looks fine, beside some minor issues on the notation (the missing definition of G, sqrt cannot be defined, it should be ##\phi^{-1}## (preimage) instead, and the equation under "subgroup" is a bit short, i.e. doesn't introduce a,b, injectivity could be a little more explicitly, i.e. why does x=y follow, resp. what properties of ##\mathbb{Q}## do you use).
 
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Thank you so much again fresh_42!
 

FAQ: Prove (Q+, *) is isomorphic to a proper subgroup of itself

1. How can you prove that (Q+, *) is isomorphic to a proper subgroup of itself?

To prove this, we need to show that there exists a bijective mapping from (Q+, *) to a proper subgroup of itself that preserves the group operation. This means that the mapping must be one-to-one, onto, and the operation must remain the same before and after the mapping.

2. What is the definition of an isomorphism in group theory?

An isomorphism is a bijective mapping between two groups that preserves the group structure, meaning the group operation is the same before and after the mapping.

3. Can (Q+, *) be isomorphic to a proper subgroup of itself?

Yes, (Q+, *) can be isomorphic to a proper subgroup of itself. This is because (Q+, *) is an infinite group, and there exists a bijective mapping from (Q+, *) to a proper subgroup of itself that preserves the group operation.

4. What is the importance of proving that (Q+, *) is isomorphic to a proper subgroup of itself?

Proving that (Q+, *) is isomorphic to a proper subgroup of itself is important because it helps us understand the structure and properties of (Q+, *). It also shows that (Q+, *) has subgroups that are isomorphic to itself, which can be useful in further exploring the group.

5. Are there any other groups that are isomorphic to (Q+, *)?

Yes, there are other groups that are isomorphic to (Q+, *). For example, the group of rational numbers with multiplication, (Q*, *), is isomorphic to (Q+, *) through the mapping f(x) = 2^x. This shows that (Q+, *) has multiple isomorphic subgroups and can be represented in different ways.

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