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8.4 Let f=(x1,x2...,x,)E S,. Show that o(f)=r.

8.4 Let f=(x1,x2...,x,)E S,. Show that o(f)=r.

Advanced Engineering Mathematics
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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**Section 8** **8.4** Let \( f = (x_1, x_2, \ldots, x_r) \in S_n \). Show that \( o(f) = r \). **Explanation**: This mathematical expression is part of a higher-level discussion on permutations within symmetric groups. Here, \( S_n \) represents the symmetric group of degree \( n \), which consists of all permutations of the set \(\{1, 2, \ldots, n\}\). The function \( f \), denoted as \( (x_1, x_2, \ldots, x_r) \), is a permutation in cycle notation. The task is to prove that the order of the permutation \( f \) (denoted as \( o(f) \)) is equal to the length of the cycle \( r \).
Transcribed Image Text:**Section 8** **8.4** Let \( f = (x_1, x_2, \ldots, x_r) \in S_n \). Show that \( o(f) = r \). **Explanation**: This mathematical expression is part of a higher-level discussion on permutations within symmetric groups. Here, \( S_n \) represents the symmetric group of degree \( n \), which consists of all permutations of the set \(\{1, 2, \ldots, n\}\). The function \( f \), denoted as \( (x_1, x_2, \ldots, x_r) \), is a permutation in cycle notation. The task is to prove that the order of the permutation \( f \) (denoted as \( o(f) \)) is equal to the length of the cycle \( r \).
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