# 3-cycle or a product of three cycles permutations

• 188818881888
In summary, the conversation discusses how to show that every element in the set of even permutations for n greater than or equal to 3 can be expressed as a 3-cycle or a product of three cycles. It is mentioned that a permutation is a bijective function from a set to itself, and for n=3 there are multiple permutations that can be expressed as 3-cycles. The conversation also touches on expressing even permutations as a product of an even number of 2-cycles and the order of A(n).
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## Homework Statement

Show that every element in A(n)= set of even permutations, for n> or equal to 3 can be expressed as a 3-cycle or a product of three cycles.

## Homework Equations

3-cycle = (_ _ _). a permutation is a function from a set A to A that is bijective.

## The Attempt at a Solution

for n=3 a permuation can be (1 2 3), (1 3 2), (2 1 3), (3 1 2) etc... need help for n>3

What factorization do you already know you can do to an even permutation?

Do you mean how can you express it? You can express an even permutation into a product of even number of 2-cycles. also the order of A(n) is n!/2

## 1. What is a 3-cycle permutation?

A 3-cycle permutation is a type of permutation in which three elements are rearranged in a specific order. This means that there are three elements involved in the permutation and they are moved to different positions in the sequence.

## 2. How is a 3-cycle permutation written?

A 3-cycle permutation is typically written using parentheses, with the elements being rearranged inside. For example, (1 2 3) represents a permutation in which element 1 is moved to the position of element 2, element 2 is moved to the position of element 3, and element 3 is moved to the position of element 1.

## 3. Can a 3-cycle permutation be reversed?

Yes, a 3-cycle permutation can be reversed by simply reversing the order of the elements inside the parentheses. For example, if the original permutation is (1 2 3), the reverse permutation would be (3 2 1).

## 4. How is a product of three cycles permutation written?

A product of three cycles permutation is written using the multiplication symbol, with each cycle being multiplied together. For example, if we have three cycles (1 2 3), (4 5 6), and (7 8 9), their product would be written as (1 2 3)(4 5 6)(7 8 9).

## 5. What is the order of a 3-cycle or a product of three cycles permutation?

The order of a 3-cycle or a product of three cycles permutation is the least common multiple of the lengths of the cycles involved. This means that the number of times the permutation must be applied to return to its original state is the least common multiple of the lengths of the cycles.

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