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Exercise 3.1.19 Show that, for n>3, the group A, is generated by 3-cycles (abc).

Exercise 3.1.19 Show that, for n>3, the group A, is generated by 3-cycles (abc).

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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3.1.19

Exercise 3.1.19 Show that, for n>3, the group A, is generated by 3-cycles (abc). Hint. We know that Sn is generated by 2-cycles. So we just need to think about prod- ucts of two 2-cycles. There are three cases to consider. (ab)(ab), (ab)(ac), (ab)(cd), where a, b, c, d are pairwise distinct integers from {1, ..., n}. For the last case, look at: (ab)(bc) (bc)(cd).
Transcribed Image Text:Exercise 3.1.19 Show that, for n>3, the group A, is generated by 3-cycles (abc). Hint. We know that Sn is generated by 2-cycles. So we just need to think about prod- ucts of two 2-cycles. There are three cases to consider. (ab)(ab), (ab)(ac), (ab)(cd), where a, b, c, d are pairwise distinct integers from {1, ..., n}. For the last case, look at: (ab)(bc) (bc)(cd).
Expert Solution
Step 1

claim- show that for n≥3 the group An is generated by 3-cycles

to prove that  An is generated by 3-cycles,  we only need to prove that π∈An

that is, 

π can be written as compositions of three cycle.

 

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