a) Prove that ~ is an equivalence relation on S5. b) Find the equivalence class of (4, 5). c) Find the equivalence class of (1, 2, 3, 4, 5). d) Determine the total number of equivalence classer

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Chapter2: Second-order Linear Odes
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please just answer a,b,c. 

**Exploring Equivalence Relations on Symmetric Groups**

**Introduction**

Consider \( S_3 \) as a subset of \( S_5 \) in the standard manner. For permutations \( \sigma, \tau \in S_5 \), the relation \( \sigma \sim \tau \) is defined if \( \sigma \tau^{-1} \in S_3 \).

**Tasks**

(a) **Prove that \( \sim \) is an equivalence relation on \( S_5 \).**

To establish that \( \sim \) is an equivalence relation, demonstrate the reflexivity, symmetry, and transitivity properties on \( S_5 \).

(b) **Find the equivalence class of \( (4, 5) \).**

Calculate the set of permutations in \( S_5 \) that relate to \( (4, 5) \) under the given equivalence relation \( \sim \).

(c) **Find the equivalence class of \( (1, 2, 3, 4, 5) \).**

Identify the permutations in \( S_5 \) equivalent to \( (1, 2, 3, 4, 5) \) using the defined equivalence relation \( \sim \).

(d) **Determine the total number of equivalence classes.**

Compute the distinct equivalence classes that cover all elements in \( S_5 \) using the relation \( \sim \).

**Conclusion**

Through these exercises, we investigate how equivalence relations can be defined using subsets of symmetric groups, showcasing their characteristics and implications in group theory.
Transcribed Image Text:**Exploring Equivalence Relations on Symmetric Groups** **Introduction** Consider \( S_3 \) as a subset of \( S_5 \) in the standard manner. For permutations \( \sigma, \tau \in S_5 \), the relation \( \sigma \sim \tau \) is defined if \( \sigma \tau^{-1} \in S_3 \). **Tasks** (a) **Prove that \( \sim \) is an equivalence relation on \( S_5 \).** To establish that \( \sim \) is an equivalence relation, demonstrate the reflexivity, symmetry, and transitivity properties on \( S_5 \). (b) **Find the equivalence class of \( (4, 5) \).** Calculate the set of permutations in \( S_5 \) that relate to \( (4, 5) \) under the given equivalence relation \( \sim \). (c) **Find the equivalence class of \( (1, 2, 3, 4, 5) \).** Identify the permutations in \( S_5 \) equivalent to \( (1, 2, 3, 4, 5) \) using the defined equivalence relation \( \sim \). (d) **Determine the total number of equivalence classes.** Compute the distinct equivalence classes that cover all elements in \( S_5 \) using the relation \( \sim \). **Conclusion** Through these exercises, we investigate how equivalence relations can be defined using subsets of symmetric groups, showcasing their characteristics and implications in group theory.
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