Write (13257)(23)(47512) as a product of disjoint cycles

In summary, the task is to write the expression (13257)(23)(47512) as a product of disjoint cycles in cycle notation. Although there may be a simpler method, currently there is no known method other than evaluating the product and writing it in cycle notation. As for finding the order of (125)(34), if the cycles are disjoint, they commute and the order is the least common multiple of both cycles' orders. Additionally, a cycle of length n is of order n.
  • #1
Mr Davis 97
1,462
44

Homework Statement


Write (13257)(23)(47512) as a product of disjoint cycles. Each bracket is a permutation of seven elements written in cycle notation.

Homework Equations

The Attempt at a Solution


This isn't too hard of a problem. One easy way would be to evaluate the entire product, and then write that product in cycle notation. However, is there an easier, faster way of doing this, just be looking at the expression (13257)(23)(47512) directly?
 
Physics news on Phys.org
  • #2
Not that I know of. And ##2## occurs in all three cycles, so any possible "rule" is likely more complicated than simply multiply them.
 
  • Like
Likes Mr Davis 97
  • #3
Also, one quick related question. I need to calculate the order of (125)(34). Is there a quick way to do this, or do I have to literally keep composing the permutation with itself until I get the identity permutation?
 
  • #4
If they are disjoint, they commute. So ##(ab)^n=a^nb^n## and the order is the least common multiple of both orders. And a cycle of length ##n## is of order ##n##.
 
  • Like
Likes Mr Davis 97

FAQ: Write (13257)(23)(47512) as a product of disjoint cycles

1. What is the meaning of "disjoint cycles"?

Disjoint cycles refer to cycles in a permutation that have no common elements. In other words, they do not share any numbers in their cycle notation.

2. Can you explain the notation used in "Write (13257)(23)(47512) as a product of disjoint cycles"?

The numbers inside the parentheses represent the elements that are being permuted. The numbers are written in a cycle notation, where the first number is mapped to the second number, the second to the third, and so on. The last number is mapped back to the first number.

3. How do I write (13257)(23)(47512) as a product of disjoint cycles?

To write (13257)(23)(47512) as a product of disjoint cycles, you need to break it down into smaller cycles. First, write down the numbers in their original order: 1234567. Then, identify the elements that are being permuted together. In this case, 1 is mapped to 3, 3 is mapped to 2, 2 is mapped to 5, 5 is mapped to 7, and 7 is mapped to 1. This forms the cycle (13752). Similarly, the cycle (23) maps 2 to 3 and 3 back to 2. The cycle (47512) maps 4 to 7, 7 to 5, 5 to 1, 1 to 4, and 2 to 2 (since it is not being permuted). Therefore, the product of disjoint cycles is (13752)(23)(47512).

4. What is the significance of writing permutations as a product of disjoint cycles?

Writing permutations as a product of disjoint cycles helps to simplify the notation and make it easier to understand. It also allows us to identify the order of the permutation, which is the smallest number of times the permutation needs to be applied to return to the original order.

5. Is there a specific order in which the cycles should be written?

There is no specific order in which the cycles should be written. However, it is common practice to write the cycles in descending order of length, starting with the longest cycle. In this case, it would be (13752)(47512)(23). But, as long as the cycles are disjoint, the order does not affect the final product.

Similar threads

Showing that an element has order 2 if product of 2-cycles
Replies
4
Views
2K
Multiplying permutation cycles
Replies
3
Views
2K
Write as a Product of Transpositions
Replies
4
Views
4K
Cycle notation of permutations
Replies
4
Views
2K
Permutations - determine order of S_n
Replies
3
Views
2K
3-cycle or a product of three cycles permutations
Replies
2
Views
2K
Product Permutations/cycle notation
Replies
1
Views
2K
Expressing as Product of Disjoint Cycles
Replies
2
Views
9K
How do I express an element in a matrix of s8 as a product of disjoint cycles?
Replies
4
Views
3K
Trying to prove that A5 has no subgroups of order 20
Replies
3
Views
6K

Hot Threads

Recent Insights

Back
Top