Product Permutations/cycle notation

In summary, to compute the product permutation qp, we start with the right-most permutation and proceed from right to left. Inside each cycle, we go from left to right. In this example, qp sends 1 to 3, 3 to 5, and 5 to 1, resulting in the product permutation (135).
  • #1
wontonsoup
1
0

Homework Statement



This is just an example in a textbook but I'm completely stumped.
This is dealing with cycles, and product permutations.
So we have
p=(341)(52)
q = (1452)

and we want to compute the product permutation qp

Homework Equations



so qp = (1452) * (341)(25)

The book writes p sends 3 to 4 and q sends 4 to 5 so qp sends 3 to 5.

The Attempt at a Solution



I know that we proceed from right to left on the permutations, and go from left to right inside the cycles.


I realize that in a cycle, say (341) 3 -> 4-> 1-> 3.

Then in this problem of finding "qp" we start with 1, and the right-most permutation. Then from there I am lost. I suppose 1 -> 3 -> ?. after 1 sends to 3 in p, do we then find what 3 would send to in q (except there is no 3 in q)?The answer for this example is (135) but I am completely lost on how to proceed. The book unfortunately isn't very clear on this. Could anybody give me some pointers on how to proceed? Thanks!
 
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  • #2
welcome to pf!

hi wontonsoup! welcome to pf! :smile:
wontonsoup said:
so qp = (1452) * (341)(25)
…
… 1 -> 3 -> ?. after 1 sends to 3 in p, do we then find what 3 would send to in q (except there is no 3 in q)?The answer for this example is (135) but I am completely lost on how to proceed.

(135) means that qp sends 1 to 3, qp sends 3 to 5, qp sends 5 to 1

qp sends 1 to 3 because (341) sends 1 to 3, and (1452) leaves 3 alone

qp sends 3 to 5 because (341) sends 3 to 4, and (1452) sends 4 to 5

qp sends 5 to 1 because (25) sends 5 to 2, and (1452) sends 2 to 1 :wink:​
 

FAQ: Product Permutations/cycle notation

1. What is the concept of product permutations/cycle notation?

Product permutations, also known as cycle notation, is a method used to represent permutations in mathematics. It involves breaking down a permutation into a series of cycles, where each cycle represents a sequence of numbers that are rearranged in a specific order.

2. How is product permutations/cycle notation useful in mathematics?

Product permutations/cycle notation is useful in mathematics because it allows us to easily represent and analyze permutations. It also helps in solving problems related to group theory and combinatorics.

3. How do you write a permutation in product permutations/cycle notation?

To write a permutation in product permutations/cycle notation, first identify the cycles by grouping the numbers that are rearranged. Then, write the cycles in a specific order, with the numbers within each cycle written in a circular form. Finally, enclose the cycles in parentheses and separate them with commas.

4. What is the difference between a cycle and a cycle notation in product permutations?

A cycle in product permutations refers to a sequence of numbers that are rearranged, while a cycle notation is the representation of a permutation in terms of cycles. In other words, a cycle is a concept, whereas cycle notation is a method of representing that concept.

5. Can you solve a permutation problem using product permutations/cycle notation?

Yes, product permutations/cycle notation can be used to solve permutation problems. It allows for easier visualization and manipulation of permutations, making it a useful tool in solving problems related to permutations, group theory, and combinatorics.

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