Hello. Needs help with disjoint cycles.

In summary, disjoint cycles refer to a set of cycles in a permutation that do not share any elements. This concept is often studied in group theory and can be used to determine the order of a permutation. It can also be used to simplify calculations and proofs involving permutations. Understanding disjoint cycles is essential for solving problems in abstract algebra and other areas of mathematics.
  • #1
Charles007
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Express as the product of disjoint cycles:
a. (1,2,3)(4,5)(1,6,7,8,9)(1,5)
b. (1,2)(1,2,3)(1,2)


I now how to do in 2 row permutations, with right to - left. can anyone tell me , how to express it without transfer it into 2 row permutations.

One more question, How do we find out it's odd or even permutaions.

Thank you. I am doing my past exam paper, university doesn't give us answer.
 
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1. What are disjoint cycles?

Disjoint cycles are a concept in mathematics that refers to cycles in a permutation that do not have any elements in common. In other words, they do not share any numbers in their cycle notation.

2. How do you represent disjoint cycles?

Disjoint cycles can be represented in cycle notation, where each cycle is enclosed in parentheses and the cycles are separated by commas. For example, (1 2 3)(4 5) denotes two disjoint cycles, one with elements 1, 2, and 3 and the other with elements 4 and 5.

3. What is the significance of disjoint cycles?

Disjoint cycles are important in mathematics, particularly in the field of group theory, because they can be used to represent a permutation in a more compact and efficient way. Additionally, they have many properties that make them useful in solving various mathematical problems.

4. How do you find the inverse of a disjoint cycle?

To find the inverse of a disjoint cycle, you simply reverse the order of the elements in each cycle. For example, the inverse of (1 2 3)(4 5) would be (3 2 1)(5 4).

5. Can disjoint cycles overlap?

No, disjoint cycles, by definition, do not overlap. They are meant to represent cycles that have no common elements, so if two cycles were to overlap, they would not be considered disjoint.

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