Proving Q+ is not Isomorphic to Q in a First Course in Algebra

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In summary, the conversation is discussing a problem in a textbook about subgroups and bijections in algebra. The first part of the problem involves showing the existence of a subgroup A of positive rationals Q+ such that the quotient group Q+/A has a specified size n. The second part involves showing that if B is a proper subgroup of rationals Q, then the quotient group Q/B has infinite size. From this, it can be concluded that Q+ is not equal to Q. The conversation then discusses the possibility of a bijection between Q+ and Q, and an isomorphism between Q+ and a subgroup A with a specific size. The conversation ends with the request for help in proving that a given function is a bijection
  • #1
Essnov
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I'm taking a first course in algebra, and in my textbook, there is the following problem:

a) Show that, for every natural # n, there is a subgroup A of Q+ such that |Q+/H| = n.

b) Suppose that B is a proper subgroup of Q. Show that |Q / B| = ∞.

c) Conclude that Q+ ≠ Q.

I did parts a and b just fine, but I'm not sure if I see how to (properly) conclude from them that Q+ ≠ Q.

I'm thinking that if there is to be a bijection from Q+ → Q, then there should be a bijection from Q+ / A → Q / B, but that's not possible since Q+ / A and Q / B should at least have the same order.

So basically if P : Q+ → Q is a bijection, the following are also bijections:
For g in Q+, h in Q,
R taking g to gA
S taking h to hB

So the composition S o P o R-1 should be a bijection from Q+ / A → Q / B, which is (I think) a contradiction, so there should be no such P.

Am I doing this properly or am I missing something more obvious?
 
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  • #2
Let [itex]f:\mathbb{Q}^+\rightarrow \mathbb{Q}[/itex] be an isomorphism. What can you say about f(A)? (where A is the subgroup with [itex]|\mathbb{Q}^+/A|=3[/itex] or another number).
 
  • #3
I cannot think of anything helpful. f(A) will be a subgroup of Q? I feel like we should have | Q / f(A) | = n but I don't know how to show this.
 
  • #4
Show that

[tex]\mathbb{Q}^+/A \rightarrow \mathbb{Q}/f(A): x+A\rightarrow f(x)+f(A)[/tex]

is a bijection. So show

1) it is well-defined
2) it is injective
3) it is surjective
 
  • #5
Thank you very much for your help :)
 

1. What does "Q+ not isomorphic to Q" mean?

"Q+" refers to the set of positive rational numbers, while "Q" refers to the set of all rational numbers. "Isomorphic" means that two sets have the same structure or properties. Therefore, "Q+ not isomorphic to Q" means that the set of positive rational numbers does not have the same structure or properties as the set of all rational numbers.

2. How can two sets have different structures or properties?

Two sets can have different structures or properties if they have different elements or if the elements are arranged differently. In the case of "Q+ not isomorphic to Q", the set of positive rational numbers only includes positive numbers, while the set of all rational numbers includes both positive and negative numbers.

3. Are there any similarities between "Q+" and "Q"?

Yes, both "Q+" and "Q" are sets of rational numbers, which means that all the numbers in both sets can be expressed as a ratio of two integers. However, their structures and properties are different, making them not isomorphic.

4. Why is it important to understand that "Q+ not isomorphic to Q"?

Understanding that "Q+ not isomorphic to Q" can help us better understand the different types of numbers and how they are related. It also helps us understand that not all sets of numbers have the same properties and that different sets can have different structures.

5. How can "Q+ not isomorphic to Q" be applied in real-life situations?

One example of how "Q+ not isomorphic to Q" can be applied in real-life is in finance and investment. The set of positive rational numbers can be used to represent positive returns on investments, while the set of all rational numbers can be used to represent both positive and negative returns. By understanding their different structures and properties, investors can make more informed decisions about their investments.

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