2. Let G = (R*, .), the multiplicative group of nonzero real numbers. Let H = {x ER: x -1} be the group with operation x * y = x+y+xy. Prove that f: G→ H, is an isomorphism, so G and H are isomorphic. f(x) = x - 1
2. Let G = (R*, .), the multiplicative group of nonzero real numbers. Let H = {x ER: x -1} be the group with operation x * y = x+y+xy. Prove that f: G→ H, is an isomorphism, so G and H are isomorphic. f(x) = x - 1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 2: Isomorphism of Groups**
Consider the group \( G = (\mathbb{R}^*, \cdot) \), which is the multiplicative group of nonzero real numbers. Define the group \( H = \{ x \in \mathbb{R} : x \neq -1 \} \) with the operation \( x * y = x + y + xy \).
**Task:**
Prove that the function
\[ f : G \rightarrow H, \quad f(x) = x - 1 \]
is an isomorphism. Therefore, demonstrate that \( G \) and \( H \) are isomorphic.
**Explanation:**
To show that \( f \) is an isomorphism, you need to verify the following:
1. **Homomorphism:** Show that the function respects the operations in the groups. That is, for \( a, b \in G \),
\[ f(a \cdot b) = f(a) * f(b) \]
2. **Bijection:** Demonstrate that \( f \) is both injective (one-to-one) and surjective (onto).
After establishing these properties, you can conclude that \( f \) is an isomorphism, therefore \( G \) and \( H \) are isomorphic groups.
**Note:** There are no graphs or additional diagrams in the original text.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa68164dd-6bba-4aa5-92bc-4824a71db092%2F1016e561-038d-4c06-94c2-7618f117751f%2F77rz55_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 2: Isomorphism of Groups**
Consider the group \( G = (\mathbb{R}^*, \cdot) \), which is the multiplicative group of nonzero real numbers. Define the group \( H = \{ x \in \mathbb{R} : x \neq -1 \} \) with the operation \( x * y = x + y + xy \).
**Task:**
Prove that the function
\[ f : G \rightarrow H, \quad f(x) = x - 1 \]
is an isomorphism. Therefore, demonstrate that \( G \) and \( H \) are isomorphic.
**Explanation:**
To show that \( f \) is an isomorphism, you need to verify the following:
1. **Homomorphism:** Show that the function respects the operations in the groups. That is, for \( a, b \in G \),
\[ f(a \cdot b) = f(a) * f(b) \]
2. **Bijection:** Demonstrate that \( f \) is both injective (one-to-one) and surjective (onto).
After establishing these properties, you can conclude that \( f \) is an isomorphism, therefore \( G \) and \( H \) are isomorphic groups.
**Note:** There are no graphs or additional diagrams in the original text.
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