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2) Let A = {1,2,...,n} and let Sn denote the permutations on A. A 2-cycle of the form (a b) is called a transposition since it transposes the elements a and b while leaving all other elements fixed. (a) Verify that (a₁ α² α3 ) = (α₁ α3)(a₁ a2) for distinct elements a1, a2, a3 € A. This shows that a 3-cycle can be expressed as a product of transpositions. (b) Show that a general k-cycle (a₁ a2 ak) in Sn can be expressed as a product of transpositions. (c) Express the following permutation in S9 as a product of transpositions (first express this as a product of cycles): f = 1 2 3 4 5 6 7 8 9 7361 4892 5

2) Let A = {1,2,...,n} and let Sn denote the permutations on A. A 2-cycle of the form (a b) is called a transposition since it transposes the elements a and b while leaving all other elements fixed. (a) Verify that (a₁ α² α3 ) = (α₁ α3)(a₁ a2) for distinct elements a1, a2, a3 € A. This shows that a 3-cycle can be expressed as a product of transpositions. (b) Show that a general k-cycle (a₁ a2 ak) in Sn can be expressed as a product of transpositions. (c) Express the following permutation in S9 as a product of transpositions (first express this as a product of cycles): f = 1 2 3 4 5 6 7 8 9 7361 4892 5

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 32EQ
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2) Let A = {1,2,...,n} and let Sn denote the permutations on A. A 2-cycle of the form (a b) is called a transposition since it transposes the elements a and b while leaving all other elements fixed. (a) Verify that (α₁ α2 α3 ) = (α₁ α3)(a₁ a2) for distinct elements a₁, a2, a3 € A. This shows that a 3-cycle can be expressed as a product of transpositions. (b) Show that a general k-cycle (a₁ a2 ak) in Sn can be expressed as a product of transpositions. (c) Express the following permutation in S9 as a product of transpositions (first express this as a product of cycles): f = 1 2 3 4 5 6 7 8 9 7 36 1 4 8925
Transcribed Image Text:2) Let A = {1,2,...,n} and let Sn denote the permutations on A. A 2-cycle of the form (a b) is called a transposition since it transposes the elements a and b while leaving all other elements fixed. (a) Verify that (α₁ α2 α3 ) = (α₁ α3)(a₁ a2) for distinct elements a₁, a2, a3 € A. This shows that a 3-cycle can be expressed as a product of transpositions. (b) Show that a general k-cycle (a₁ a2 ak) in Sn can be expressed as a product of transpositions. (c) Express the following permutation in S9 as a product of transpositions (first express this as a product of cycles): f = 1 2 3 4 5 6 7 8 9 7 36 1 4 8925
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