Homepage › Solution manuals › David S. Dummit › Abstract Algebra › Exercise 1.3.2 (Cycle decompositions 2)

Abstract Algebra

Exercise 1.3.2 (Cycle decompositions 2)

Let σ be the permutation

σ = ( 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 13 2 15 14 10 6 12 3 4 1 7 9 5 11 8 )

and let τ be the permutation

τ = ( 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 14 9 10 2 12 6 5 11 15 3 8 7 4 1 13 )

Find the cycle decompositions of the following permutations: σ , τ , σ 2 , στ , τσ , and τ 2 σ .

Answers

Proof. Using the Cycle Decomposition Algorithm, we obtain

σ = ( 1 13 5 10 ) ( 3 15 8 ) ( 4 14 11 7 12 9 ) τ = ( 1 14 ) ( 2 9 15 13 4 ) ( 3 10 ) ( 5 12 7 ) ( 8 11 ) .

Since the disjoint cycles commute,

σ 2 = ( 1 13 5 10 ) 2 ( 3 15 8 ) 2 ( 4 14 11 7 12 9 ) 2 = ( 1 5 ) ( 13 10 ) ( 3 8 15 ) ( 4 11 12 ) ( 7 9 14 ) , τ 2 = ( 1 14 ) 2 ( 2 9 15 13 4 ) 2 ( 3 10 ) 2 ( 5 12 7 ) 2 ( 8 11 ) 2 = ( 2 15 4 9 13 ) ( 5 7 12 ) .

Thus

τ 2 = ( 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 15 3 9 7 6 12 8 13 10 11 5 2 14 4 )

Moreover

στ = ( 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 11 4 1 2 9 6 10 7 8 15 3 12 14 13 5 ) = ( 1 11 3 ) ( 2 4 ) ( 5 9 8 7 10 15 ) ( 13 14 ) ,

τσ = ( 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 4 9 13 1 3 6 7 10 2 14 5 15 12 8 11 ) = ( 1 4 ) ( 2 9 ) ( 3 13 12 15 11 5 ) ( 8 10 14 ) , τ 2 σ = ( 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 15 4 14 10 6 5 3 9 1 12 13 7 11 8 ) = ( 1 2 15 8 3 4 14 11 12 13 7 5 10 ) .

Verifications with Sagemath

(the usual product * is from the composition from left to write, so I used “sigma.left_action(tau)” to compute στ )

sage: sigma = Permutation([13, 2, 15, 14, 10, 6, 12, 3, 4, 1, 7, 9, 5, 11, 8]); sigma
[13, 2, 15, 14, 10, 6, 12, 3, 4, 1, 7, 9, 5, 11, 8]
sage: tau = Permutation([14, 9, 10, 2, 12, 6, 5, 11, 15, 3, 8, 7, 4, 1, 13])
sage: tau2 = tau * tau; tau2
[1, 15, 3, 9, 7, 6, 12, 8, 13, 10, 11, 5, 2, 14, 4]
sage: sigma2 = sigma * sigma
sage: sigma_tau = sigma.left_action_product(tau); sigma_tau
[11, 4, 1, 2, 9, 6, 10, 7, 8, 15, 3, 12, 14, 13, 5]
sage: tau_sigma = tau.left_action_product(sigma); tau_sigma
[4, 9, 13, 1, 3, 6, 7, 10, 2, 14, 5, 15, 12, 8, 11]
sage: sigma.cycle_string()
’(1,13,5,10)(3,15,8)(4,14,11,7,12,9)’
sage: tau.cycle_string()
’(1,14)(2,9,15,13,4)(3,10)(5,12,7)(8,11)’
sage: sigma_tau.cycle_string()
’(1,11,3)(2,4)(5,9,8,7,10,15)(13,14)’
sage: tau_sigma.cycle_string()
’(1,4)(2,9)(3,13,12,15,11,5)(8,10,14)’
sage: eta = tau2.left_action_product(sigma); eta
[2, 15, 4, 14, 10, 6, 5, 3, 9, 1, 12, 13, 7, 11, 8]
sage: eta.cycle_string()
’(1,2,15,8,3,4,14,11,12,13,7,5,10)’

This confirms our results. □

User profile picture
2025-09-11 11:07
Comments