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Exercise 8.3. a) Show that if o is a k-cycle, then ok is the identity.

Exercise 8.3. a) Show that if o is a k-cycle, then ok is the identity.

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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This is group theory question, please explain it using simple words step by step, thanks :)

Exercise 8.3. a) Show that if o is a k-cycle, then ok is the identity. b) Show that if o is a written as a product of disjoint cycles 0₁,...,01 of lengths 11,..., lk. Then oll2 is the identity. c) Suppose o € Sym(n) is a k-cycle. Show that 707¹ is also a k-cycle for every 7 € Sym(n). More precisely, show that if σ = (a₁ a2₂ ….. ak), then TƠ7¯¹ = (7(α₁) 7(α2) ….. T(ak)). TOT-¹ o
Transcribed Image Text:Exercise 8.3. a) Show that if o is a k-cycle, then ok is the identity. b) Show that if o is a written as a product of disjoint cycles 0₁,...,01 of lengths 11,..., lk. Then oll2 is the identity. c) Suppose o € Sym(n) is a k-cycle. Show that 707¹ is also a k-cycle for every 7 € Sym(n). More precisely, show that if σ = (a₁ a2₂ ….. ak), then TƠ7¯¹ = (7(α₁) 7(α2) ….. T(ak)). TOT-¹ o
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