Exercise 14.4.28. Express the following permutations as products of trans- positions. (a) (14356) (d) (17254)(1423)(154632) (b) (156)(234) (e) (142637)(2359) (c) (1426)(142) (f) (13579)(2468)(19753)(2864)

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.1: Finite Permutation Groups
Problem 8E: Express each permutation in Exercise 2 as a product of transpositions. Express each permutation as a...
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Please do Exercise 14.4.28 part D and F and please show step by step and explain

Recall that we also know that any permutation can be written as a product
of disjoint cycles, which leads to:
Proposition 14.4.27. Any permutation of a finite set containing at least
two elements can be written as the product of transpositions.
PROOF. First write the permutation as a product of cycles: then write each
cycle as a product of transpositions.
Exercise 14.4.28. Express the following permutations as products of trans-
positions.
(a) (14356)
(d) (17254)(1423)(154632)
(b) (156)(234)
(e) (142637)(2359)
(c) (1426)(142)
(f) (13579)(2468)(19753)(2864)
Even the identity permutation id can be expressed as the product of trans-
positions:
Transcribed Image Text:Recall that we also know that any permutation can be written as a product of disjoint cycles, which leads to: Proposition 14.4.27. Any permutation of a finite set containing at least two elements can be written as the product of transpositions. PROOF. First write the permutation as a product of cycles: then write each cycle as a product of transpositions. Exercise 14.4.28. Express the following permutations as products of trans- positions. (a) (14356) (d) (17254)(1423)(154632) (b) (156)(234) (e) (142637)(2359) (c) (1426)(142) (f) (13579)(2468)(19753)(2864) Even the identity permutation id can be expressed as the product of trans- positions:
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