Are pq and qp always 3-cycles?

[Brucezhou]

Homework Statement

In the text, the products pq and qp of the permutations (2 3 4) and (1 3 5) were seen to be different. However, both products turned out to be 3-cycles. Is this an accident?

Homework Equations

p=(3 4 1)(2 5)
q=(1 4 5 2)
where p and q are permutations

The Attempt at a Solution

Based on many examples I made, this is obviously not an accident. For example, if the product is equal to (1 3 4)(5 2)(6), then the other product will also in the form (a b c)(d f)(e). However, I cannot come up with a general way to prove the truth. I have tried to find any relationship between the number of fixed numbers(the number just moves to itself after permutation, such as "6" in the previous example) and the product, but it doesn't show any regality. The truth is that if a number is fixed in both permutations, then it must be fixed in the products.

 

[Dick]

Homework Statement

In the text, the products pq and qp of the permutations (2 3 4) and (1 3 5) were seen to be different. However, both products turned out to be 3-cycles. Is this an accident?

Homework Equations

p=(3 4 1)(2 5)
q=(1 4 5 2)
where p and q are permutations

The Attempt at a Solution

Based on many examples I made, this is obviously not an accident. For example, if the product is equal to (1 3 4)(5 2)(6), then the other product will also in the form (a b c)(d f)(e). However, I cannot come up with a general way to prove the truth. I have tried to find any relationship between the number of fixed numbers(the number just moves to itself after permutation, such as "6" in the previous example) and the product, but it doesn't show any regality. The truth is that if a number is fixed in both permutations, then it must be fixed in the products.

No, it's not an accident. Do you know any group theory? If two permutations are conjugate then they have the same cycle structure. Can you show pq and qp are conjugate?

 

[Brucezhou]

No, it's not an accident. Do you know any group theory? If two permutation are conjugate then they have the same cycle structure. Can you show pq and qp are conjugate?

Group is in the next chapter. Thank you very much. I am learning that chapter.