Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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

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
See similar textbooks
icon
Related questions
Question
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
Expert Solution
steps

Step by step

Solved in 4 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS