Expressing as Product of Disjoint Cycles

• apalmer3
In summary, the concept of "Expressing as Product of Disjoint Cycles" is a method used in mathematics, specifically in group theory, to represent a permutation as a product of smaller, simpler cycles. This is done by breaking down the permutation into smaller cycles and then combining them into one product. It is useful for understanding the structure of the permutation and performing calculations, and is related to other mathematical concepts such as group theory and abstract algebra.
apalmer3

Homework Statement

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)

The Attempt at a Solution

a. (1, 2, 3) ( 4, 5, 6, 7, 8, 9)
b. (1, 2, 3)

These are the answers I got directly through the product. Would anybody be willing to check those answers? Thank you!

Sorry. I looked at it again. Here's my second attempt at a solution:

a. (1,4,5,6,7,8,9,2,3)
b. (1,3,2)

It looks fine to me, if you are applying the permutations from right to left.

1. What is the concept of "Expressing as Product of Disjoint Cycles"?

The concept of "Expressing as Product of Disjoint Cycles" is a method used in mathematics, specifically in the field of group theory, to represent a permutation as a product of smaller, simpler cycles. A permutation is a rearrangement of a set of elements, and this method helps to break down a complex permutation into smaller, more manageable parts.

2. How is a permutation expressed as a product of disjoint cycles?

To express a permutation as a product of disjoint cycles, we first write out the permutation in cycle notation, with the elements being moved in each cycle listed within parentheses. Then, we can break down the permutation into smaller cycles by grouping elements that are moved together. Finally, we can combine these smaller cycles into one product, with each cycle being separated by a dot.

3. Why is it useful to express a permutation as a product of disjoint cycles?

Expressing a permutation as a product of disjoint cycles can help us understand the structure of the permutation better. It also allows us to perform calculations and operations on the permutation more easily, as we can break it down into smaller parts. Additionally, this method can be used to prove certain properties and theorems about permutations.

4. Can a permutation be expressed as a product of non-disjoint cycles?

Yes, a permutation can be expressed as a product of non-disjoint cycles. However, the product of disjoint cycles is considered to be a more simplified and efficient representation of the permutation.

5. How is "Expressing as Product of Disjoint Cycles" related to other mathematical concepts?

"Expressing as Product of Disjoint Cycles" is closely related to the concept of group theory, as it is a method used to analyze and understand the structure of permutations within a group. It is also related to other concepts in abstract algebra, such as permutations, cycles, and subgroup decompositions.

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