Multiplicative groups of nonzero reals and pos. reals

  • Thread starter Bachelier
  • Start date
  • Tags
    Groups
In summary, the conversation discusses the isomorphism between the quotient groups \mathbb R^*/N and \mathbb R^{**}, where N is the interval (-1,1). The first isomorphism theorem is used to prove that \mathbb R^*/\mathbb R^{**} is isomorphic to \mathbb Z/2\mathbb Z, with an embedding given by the homomorphism \psi:\mathbb R^*\rightarrow\ N. The need for an embedding rather than just a surjection is emphasized.
  • #1
Bachelier
376
0
WTS is that [itex]\mathbb R^*/N \ \cong \ \mathbb R^{**}[/itex] where [itex]N = (-1, 1)[/itex]

then prove that [itex]\mathbb R^*/\mathbb R^{**} \ is \ \cong \ to \mathbb Z/2\mathbb Z[/itex]

So the best answer in my opinion is to construct a surjection and use the first iso thm.

[itex]f:\mathbb R^*\rightarrow\mathbb R^{**}[/itex]

[tex]f(x)=|x|,[/tex] is onto by construction. clearly a homomorphism

[itex]Ker \ (f) = N[/itex], hence [itex]\mathbb R^*/N \ \cong \ \mathbb R^{**}[/itex]

part 2


[itex]ψ:\mathbb R^*\rightarrow\ \ N[/itex]

[tex]ψ(x)=1 \ if \ x>0 \ and \ ψ(x)=-1 \ if \ x<0[/tex]

by same thm, [itex]\mathbb R^*/\mathbb R^{**} \cong \mathbb Z/2\mathbb Z[/itex]

because it has 2 elements one of each is the identity.
 
Physics news on Phys.org
  • #2
As long as you do not tell us how you imbed ##\mathbb{R}^{**}## in ##\mathbb{R}^*## as a normal subgroup we cannot say anything. We need an embedding, not a surjection.
 

1. What is a multiplicative group of nonzero real numbers?

A multiplicative group of nonzero real numbers is a set of real numbers that can be multiplied together and still remain within the set. In other words, the set is closed under multiplication. The identity element in this group is 1, and every element in the group has an inverse, meaning it can be divided by another element in the group to equal 1.

2. How is a multiplicative group of nonzero real numbers different from a regular group?

A multiplicative group is different from a regular group in that the operation used is multiplication instead of addition. This means that the identity element is 1 instead of 0, and the inverse of an element is found by dividing instead of subtracting. Additionally, the notation used for a multiplicative group is typically written as multiplcative notation, such as (ℝ*, x) instead of additive notation.

3. What is the order of a multiplicative group of nonzero real numbers?

The order of a multiplicative group of nonzero real numbers is the number of elements in the group. This is equal to the number of positive real numbers less than or equal to the group's largest element. For example, the multiplicative group (ℝ*, x) has an infinite order since there are an infinite number of positive real numbers.

4. Can the set of positive real numbers be considered a multiplicative group?

Yes, the set of positive real numbers (ℝ+, x) can be considered a multiplicative group. It is closed under multiplication, has 1 as the identity element, and every element has an inverse (the reciprocal). However, it is not considered a multiplicative group of nonzero real numbers since it does not include 0, which is necessary for a group to be considered non-zero.

5. What is the significance of studying multiplicative groups of nonzero real numbers?

Studying multiplicative groups of nonzero real numbers can help us better understand the properties and relationships of real numbers. It also has many applications in fields such as number theory, algebra, and cryptography. Additionally, understanding multiplicative groups can help us solve problems involving ratios, proportions, and exponential growth.

Similar threads

  • Linear and Abstract Algebra
Replies
2
Views
947
  • Linear and Abstract Algebra
2
Replies
52
Views
2K
  • Linear and Abstract Algebra
Replies
4
Views
911
  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
774
  • Linear and Abstract Algebra
Replies
9
Views
852
  • Linear and Abstract Algebra
Replies
1
Views
935
  • Linear and Abstract Algebra
Replies
17
Views
1K
  • Linear and Abstract Algebra
Replies
19
Views
4K
  • Linear and Abstract Algebra
Replies
7
Views
1K
Back
Top