Let ℝ∗
be the group of nonzero real numbers with multiplication, and let
𝐻 = {𝑥 ∈ ℝ∗
| 𝑥
𝑛
∈ ℚ for some 𝑛 ∈ ℕ>0}.
1. Show that 𝐻 is a subgroup of ℝ∗
, and that it is normal.
2. Let 𝑥𝐻 be an element of the quotient group ℝ∗/𝐻. Show that if 𝑥𝐻 has finite
order, then it must be the neutral element 1𝐻.

Let's solve the problem step by step.

### 1. Show that \( H \) is a subgroup of \( \mathbb{R}^* \) and that it is normal.

#### Subgroup Criteria:
To show that \( H \) is a subgroup of \( \mathbb{R}^* \), we need to verify the following three criteria:
1. **Closure**: If \( x, y \in H \), then \( xy \in H \).
2. **Identity**: The identity element \( 1 \in H \).
3. **Inverses**: If \( x \in H \), then \( x^{-1} \in H \).

Let's analyze each criterion.

1. **Closure**:  
   Suppose \( x \in H \) and \( y \in H \). By definition of \( H \), there exist positive integers \( m \) and \( n \) such that \( x^m \in \mathbb{Q} \) and \( y^n \in \mathbb{Q} \). We need to show that \( xy \in H \), i.e., that \( (xy)^k \in \mathbb{Q} \) for some \( k \in \mathbb{N} \).

   Consider the product \( (xy)^{mn} = x^{mn}y^{mn} \). Since \( x^m \in \mathbb{Q} \) and \( y^n \in \mathbb{Q} \), it follows that \( x^{mn} \in \mathbb{Q} \) and \( y^{mn} \in \mathbb{Q} \). Therefore, \( (xy)^{mn} = x^{mn} y^{mn} \in \mathbb{Q} \). Thus, \( xy \in H \), proving closure.

2. **Identity**:  
   The identity element in \( \mathbb{R}^* \) under multiplication is \( 1 \). Clearly, \( 1^n = 1 \in \mathbb{Q} \) for any positive integer \( n \), so \( 1 \in H \).

3. **Inverses**:  
   If \( x \in H \), then there exists some positive integer \( n \) such that \( x^n \in \mathbb{Q} \). Consider the inverse \( x^{-1} \in \mathbb{R}^* \). We need to show that \( x^{-1} \in H \).

   Note that \( (x^{-1})^n = (x^n)^{-1} \). Since \( x^n \in \mathbb{Q} \) and the rationals \( \mathbb{Q} \) are closed under taking inverses (except for zero, but \( x^n \neq 0 \) since \( x \in \mathbb{R}^* \)), we have \( (x^n)^{-1} \in \mathbb{Q} \). Therefore, \( x^{-1} \in H \).

Since \( H \) satisfies all three subgroup criteria, \( H \) is a subgroup of \( \mathbb{R}^* \).

#### Normality:
To show that \( H \) is a normal subgroup of \( \mathbb{R}^* \), we need to verify that for any \( x \in \mathbb{R}^* \) and \( h \in H \), the conjugate \( xhx^{-1} \) is in \( H \).

Given \( h \in H \), there exists some positive integer \( n \) such that \( h^n \in \mathbb{Q} \). Consider the element \( xhx^{-1} \). We need to show that \( (xhx^{-1})^m \in \mathbb{Q} \) for some positive integer \( m \).

Calculate:
\[
(xhx^{-1})^m = xh^m x^{-1}
\]
Since \( h^m \in \mathbb{Q} \), \( xh^m x^{-1} \) is just a conjugate of a rational number, and since the rationals are closed under conjugation by nonzero real numbers (i.e., \( x q x^{-1} = q \) for any \( q \in \mathbb{Q} \) and \( x \in \mathbb{R}^* \)), we have \( xh^m x^{-1} \in \mathbb{Q} \). Therefore, \( xhx^{-1} \in H \), showing that \( H \) is normal.

### 2. Show that if \( xH \) has finite order in the quotient group \( \mathbb{R}^*/H \), then it must be the neutral element \( 1H \).

Suppose \( xH \) has finite order in the quotient group \( \mathbb{R}^*/H \). This means there exists some positive integer \( k \) such that \( (xH)^k = H \).

This implies:
\[
x^k H = H
\]
which further implies:
\[
x^k \in H
\]
Since \( x^k \in H \), by the definition of \( H \), there exists a positive integer \( n \) such that \( (x^k)^n \in \mathbb{Q} \). But this means \( x^{kn} \in \mathbb{Q} \), so \( x \in H \). Therefore, \( xH = H \), which is the neutral element in the quotient group \( \mathbb{R}^*/H \).

This concludes the proof that if \( xH \) has finite order, then it must be the neutral element \( 1H \).

---

Would you like a more detailed explanation of any part? Here are 8 follow-up questions to expand on this topic:

1. What are the implications of \( H \) being a normal subgroup in \( \mathbb{R}^* \)?
2. How does the concept of closure apply in more complex groups?
3. Can you provide examples of other normal subgroups in well-known groups?
4. How would the proof change if we were dealing with addition instead of multiplication?
5. What is the significance of the quotient group \( \mathbb{R}^*/H \)?
6. How does this result relate to the concept of torsion elements in group theory?
7. What are the possible orders of elements in the group \( \mathbb{R}^*/H \)?
8. Could \( H \) be characterized differently to explore other interesting properties?

**Tip**: When analyzing group structures, it's often helpful to visualize the operations and relations using specific examples or simpler cases to build intuition.

Learn about the subgroup 𝐻 of nonzero real numbers 𝑅∗ and its properties. Discover why 𝐻 is a subgroup and a normal subgroup in 𝑅∗, and explore the concept of quotient groups. Detailed proofs and explanations provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation