27. Let X and Y be sets having the same cardinality, and suppose that 0:X Y is a bijection. Define 0: Sym(X) + Sym(Y) by (0) = 0.0.0-1. Prove that o defines an isomorphism of groups.

We are tasked with proving that the function \( \theta: \text{Sym}(X) \to \text{Sym}(Y) \), defined by \( \theta(\sigma) = \varphi \circ \sigma \circ \varphi^{-1} \), is an isomorphism of groups, where \( \varphi: X \to Y \) is a bijection and \( \text{Sym}(X) \) and \( \text{Sym}(Y) \) represent the symmetric groups on sets \( X \) and \( Y \) respectively.

### Steps of the proof:

#### 1. **Define the function \(\theta\)**:
   - We are given a bijection \( \varphi: X \to Y \).
   - Define \( \theta: \text{Sym}(X) \to \text{Sym}(Y) \) by \( \theta(\sigma) = \varphi \circ \sigma \circ \varphi^{-1} \) for all \( \sigma \in \text{Sym}(X) \).
   - \( \sigma \in \text{Sym}(X) \) is a permutation on the set \( X \), and \( \theta(\sigma) \in \text{Sym}(Y) \) is a permutation on \( Y \).

#### 2. **Show that \(\theta\) is a homomorphism**:
   To show that \( \theta \) is a homomorphism, we need to verify that for all \( \sigma_1, \sigma_2 \in \text{Sym}(X) \),
   \[
   \theta(\sigma_1 \circ \sigma_2) = \theta(\sigma_1) \circ \theta(\sigma_2).
   \]

   - Left-hand side: \( \theta(\sigma_1 \circ \sigma_2) = \varphi \circ (\sigma_1 \circ \sigma_2) \circ \varphi^{-1} \).
   - Right-hand side: \( \theta(\sigma_1) \circ \theta(\sigma_2) = (\varphi \circ \sigma_1 \circ \varphi^{-1}) \circ (\varphi \circ \sigma_2 \circ \varphi^{-1}) \).
   
   Now, simplify the right-hand side:
   \[
   (\varphi \circ \sigma_1 \circ \varphi^{-1}) \circ (\varphi \circ \sigma_2 \circ \varphi^{-1}) = \varphi \circ \sigma_1 \circ (\varphi^{-1} \circ \varphi) \circ \sigma_2 \circ \varphi^{-1} = \varphi \circ \sigma_1 \circ \sigma_2 \circ \varphi^{-1}.
   \]
   Since the left-hand side and right-hand side are equal, \( \theta \) is a homomorphism.

#### 3. **Show that \(\theta\) is injective**:
   To show injectivity, assume \( \theta(\sigma_1) = \theta(\sigma_2) \). This means:
   \[
   \varphi \circ \sigma_1 \circ \varphi^{-1} = \varphi \circ \sigma_2 \circ \varphi^{-1}.
   \]
   Applying \( \varphi^{-1} \) from the left and \( \varphi \) from the right, we get:
   \[
   \sigma_1 = \sigma_2.
   \]
   Therefore, \( \theta \) is injective.

#### 4. **Show that \(\theta\) is surjective**:
   Let \( \tau \in \text{Sym}(Y) \). We need to find \( \sigma \in \text{Sym}(X) \) such that \( \theta(\sigma) = \tau \), i.e., \( \varphi \circ \sigma \circ \varphi^{-1} = \tau \).
   Define \( \sigma = \varphi^{-1} \circ \tau \circ \varphi \). Then,
   \[
   \theta(\sigma) = \varphi \circ (\varphi^{-1} \circ \tau \circ \varphi) \circ \varphi^{-1} = \tau.
   \]
   Therefore, \( \theta \) is surjective.

#### 5. **Conclusion: \(\theta\) is an isomorphism**:
   Since \( \theta \) is both injective and surjective, it is a bijection. Moreover, since \( \theta \) is a homomorphism, it is an isomorphism of groups.

Thus, \( \theta: \text{Sym}(X) \to \text{Sym}(Y) \) defines an isomorphism of groups.

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Do you want more details on any of the steps or have any questions?

Here are 5 related questions to explore:
1. What is a group homomorphism, and how is it different from an isomorphism?
2. How do symmetric groups generalize to non-finite sets?
3. What are automorphisms of a group, and how are they related to this problem?
4. How can we generalize this proof to other group actions beyond symmetric groups?
5. Can we define a similar isomorphism if \( X \) and \( Y \) have different cardinalities?

**Tip**: When proving that a function is a group isomorphism, you must show it is both a homomorphism and a bijection (injective and surjective).

Let X and Y be sets having the same cardinality, and suppose that ϕ: X → Y is a bijection. Define ϕ: Sym(X) → Sym(Y) by θ(σ) = ϕ ◦ σ ◦ ϕ⁻¹. Prove that θ defines an isomorphism of groups.

This detailed solution proves that the function θ: Sym(X) → Sym(Y), defined by θ(σ) = ϕ ◦ σ ◦ ϕ⁻¹, is a group isomorphism. The proof utilizes fundamental concepts from group theory, including bijections, homomorphisms, and the group isomorphism theorem. Suitable for undergraduate students studying abstract algebra.

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