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.


### 1. Showing that \( H \) is a Subgroup of \( \mathbb{R}^* \) and that it is Normal

First, let's restate the definition of \( H \):
\[ H = \{ x \in \mathbb{R}^* \mid x^n \in \mathbb{Q} \text{ for some } n \in \mathbb{N} \}. \]

To show that \( H \) is a subgroup of \( \mathbb{R}^* \), we need to verify the subgroup criteria:
1. The identity element of \( \mathbb{R}^* \) is in \( H \).
2. \( H \) is closed under the group operation (multiplication).
3. \( H \) is closed under taking inverses.

**Identity Element:**
The identity element in \( \mathbb{R}^* \) is \( 1 \). We need to check if \( 1 \in H \).
\[ 1^n = 1 \in \mathbb{Q} \text{ for any } n \in \mathbb{N}. \]
Thus, \( 1 \in H \).

**Closure under Group Operation:**
Let \( x, y \in H \). This means there exist \( n, m \in \mathbb{N} \) such that
\[ x^n \in \mathbb{Q} \quad \text{and} \quad y^m \in \mathbb{Q}. \]
We need to show that \( xy \in H \). Consider \( (xy)^{nm} \):
\[ (xy)^{nm} = (x^n)^m (y^m)^n \in \mathbb{Q} \cdot \mathbb{Q} = \mathbb{Q}. \]
Thus, \( xy \in H \).

**Closure under Inverses:**
Let \( x \in H \). This means there exists \( n \in \mathbb{N} \) such that \( x^n \in \mathbb{Q} \). We need to show that \( x^{-1} \in H \). Consider \( (x^{-1})^n \):
\[ (x^{-1})^n = (x^n)^{-1}. \]
Since \( x^n \in \mathbb{Q} \) and the inverse of a nonzero rational number is also a rational number,
\[ (x^n)^{-1} \in \mathbb{Q}. \]
Thus, \( x^{-1} \in H \).

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

**Normality:**
To show that \( H \) is a normal subgroup of \( \mathbb{R}^* \), we need to show that for all \( x \in \mathbb{R}^* \) and \( h \in H \),
\[ xhx^{-1} \in H. \]
Let \( h \in H \). This means there exists \( n \in \mathbb{N} \) such that \( h^n \in \mathbb{Q} \). We need to show that \( xhx^{-1} \in H \). Consider \( (xhx^{-1})^n \):
\[ (xhx^{-1})^n = xh^n x^{-1}. \]
Since \( h^n \in \mathbb{Q} \), and \( \mathbb{Q} \) is closed under multiplication by any nonzero real number and its inverse,
\[ xh^n x^{-1} \in \mathbb{Q}. \]
Thus, \( xhx^{-1} \in H \), proving that \( H \) is normal in \( \mathbb{R}^* \).

### 2. Showing that If \( xH \) has Finite Order, then \( xH \) must be the Neutral Element \( 1H \)

Let \( xH \) be an element of the quotient group \( \mathbb{R}^*/H \). Suppose \( xH \) has finite order, say \( n \), meaning
\[ (xH)^n = H. \]
This implies
\[ x^nH = H, \]
which means
\[ x^n \in H. \]

By the definition of \( H \), there exists some \( m \in \mathbb{N} \) such that \( (x^n)^m \in \mathbb{Q} \). Thus,
\[ x^{nm} \in \mathbb{Q}. \]

Consider the order of \( xH \). If \( xH \) has finite order \( n \), then \( x^{nm} \in \mathbb{Q} \). Since any nonzero rational number is an element of \( H \), we have \( x^{nm} \in H \). Therefore,
\[ x^{nm}H = H, \]
which implies
\[ xH = H. \]

Hence, \( xH = 1H \), the neutral element in the quotient group \( \mathbb{R}^*/H \).

### Summary

1. \( H = \{ x \in \mathbb{R}^* \mid x^n \in \mathbb{Q} \text{ for some } n \in \mathbb{N} \} \) is a subgroup of \( \mathbb{R}^* \), and \( H \) is normal in \( \mathbb{R}^* \).
2. If \( xH \) in the quotient group \( \mathbb{R}^*/H \) has finite order, then \( xH \) must be the neutral element \( 1H \).

Would you like more details or have any questions? Here are some related questions:

1. What is a quotient group, and how is it defined?
2. Why is the closure under inverses important for proving subgroup properties?
3. Can you provide an example of a normal subgroup in a different group?
4. What are the implications of a subgroup being normal in terms of group structure?
5. How does the order of an element in a quotient group relate to its properties in the original group?
6. Why is \( \mathbb{Q} \) being closed under multiplication important for the proof?
7. Can you explain the relationship between rational numbers and the subgroup \( H \)?
8. What are some applications of normal subgroups and quotient groups in other areas of mathematics?

**Tip:** When proving properties about subgroups and quotient groups, always refer to the definitions and use fundamental properties of group operations to guide your proof.

Explore the mathematical proof that 𝐻, defined as the set of nonzero real numbers 𝑥 such that 𝑥ⁿ ∈ ℚ for some 𝑛 ∈ ℕ>0, forms a subgroup of ℝ∗ and is normal. Learn why if 𝑥𝐻 has finite order in the quotient group ℝ∗/𝐻, it must be the neutral element 1𝐻.

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