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Let e € Sn be the identity. Assume that e = B1B2. Br, where B₁, B2,... Br are 2-cycles. Choose the correct option from below and explain why. (a) r is always odd. (b) r is always even. (c) r can be odd or even, it depends on the specific decomposition that we use.

Let e € Sn be the identity. Assume that e = B1B2. Br, where B₁, B2,... Br are 2-cycles. Choose the correct option from below and explain why. (a) r is always odd. (b) r is always even. (c) r can be odd or even, it depends on the specific decomposition that we use.

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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**Text Transcription:** Let \( e \in S_n \) be the identity. Assume that \( e = \beta_1 \beta_2 \ldots \beta_r \), where \( \beta_1, \beta_2, \ldots, \beta_r \) are 2-cycles. Choose the correct option from below and explain why: (a) \( r \) is always odd. (b) \( r \) is always even. (c) \( r \) can be odd or even, it depends on the specific decomposition that we use.
Transcribed Image Text:**Text Transcription:** Let \( e \in S_n \) be the identity. Assume that \( e = \beta_1 \beta_2 \ldots \beta_r \), where \( \beta_1, \beta_2, \ldots, \beta_r \) are 2-cycles. Choose the correct option from below and explain why: (a) \( r \) is always odd. (b) \( r \) is always even. (c) \( r \) can be odd or even, it depends on the specific decomposition that we use.
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