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Exercise 55. Write all elements of S3 and S4 and identify the subgroups A and A4.

Exercise 55. Write all elements of S3 and S4 and identify the subgroups A and A4.

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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2.3. CONJUGACY CLASSES OF S AND An 33 Exercise 55. Write all elements of S3 and S4 and identify the subgroups A and A4 Exercise 56. Prove that An is generated by permutations of type (12k), where 3 ≤ k ≤n. Exercise 57. Prove that the dihedral group De is isomorphic to the Sym- metric group S3. Exercise 58. Prove that Z(Sn) = 1. 2.3 Conjugacy classes of Sn and An By the results of the previous section, every permutation is a product of some cycles. The cyclic structure of a permutation shows the number of the cycles and their orders. For example, if o = (123) (45) (67,8,9), is a permutation in S12, then the cyclic structure of o is 132¹3¹4¹, which means o has 3 cycles of order 1, and on cycle of order2, one cycle of order 3
Transcribed Image Text:2.3. CONJUGACY CLASSES OF S AND An 33 Exercise 55. Write all elements of S3 and S4 and identify the subgroups A and A4 Exercise 56. Prove that An is generated by permutations of type (12k), where 3 ≤ k ≤n. Exercise 57. Prove that the dihedral group De is isomorphic to the Sym- metric group S3. Exercise 58. Prove that Z(Sn) = 1. 2.3 Conjugacy classes of Sn and An By the results of the previous section, every permutation is a product of some cycles. The cyclic structure of a permutation shows the number of the cycles and their orders. For example, if o = (123) (45) (67,8,9), is a permutation in S12, then the cyclic structure of o is 132¹3¹4¹, which means o has 3 cycles of order 1, and on cycle of order2, one cycle of order 3
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