Algebra and Permutations: Determining the Parity of an Element in S_n

In summary, the conversation discusses whether an element σ of Sn, which satisfies σ5=1, is necessarily odd or even. The equation Parity(σ)=(∑i=1k(|ci|−1))mod2 is mentioned as a potential solution. It is suggested to use the fact that every permutation can be decomposed into a product of disjoint or 2-cycles to determine the parity of σ. The solution is then hinted at by pointing out that 1 is an even permutation and the implications of σ5 being even.
  • #1
decerto
87
2

Homework Statement



Suppose σ is an element of S_n, if σ^5=1, is σ necessarily odd or even

Homework Equations



[itex]Parity(\sigma)= (\sum_{i=1}^k(|c_i|-1)) mod 2[/itex]

The Attempt at a Solution



Really unsure how to even start this, I think I have to use the fact you can decompose every permutation into a product of disjoint or 2 cycles and use their properties to show in what cases it's even or odd but I'm not really sure.
 
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  • #2
decerto said:

Homework Statement



Suppose σ is an element of S_n, if σ^5=1, is σ necessarily odd or even

Homework Equations



[itex]Parity(\sigma)= (\sum_{i=1}^k(|c_i|-1)) mod 2[/itex]

The Attempt at a Solution



Really unsure how to even start this, I think I have to use the fact you can decompose every permutation into a product of disjoint or 2 cycles and use their properties to show in what cases it's even or odd but I'm not really sure.

##\sigma^5=\sigma \sigma \sigma \sigma \sigma=1##. ##1## is an even permutation. Suppose ##\sigma## were odd. Then what would be the parity of ##\sigma^5##?
 
  • #3
What does σ5 being even tell you?
 
Last edited:

1. What is algebra?

Algebra is a branch of mathematics that deals with the study of symbols and the rules for manipulating them to solve problems.

2. What are the basic principles of algebra?

The basic principles of algebra include the use of variables, constants, and operations such as addition, subtraction, multiplication, and division to solve equations and manipulate expressions.

3. What is a permutation in algebra?

In algebra, a permutation is an arrangement of a set of objects in a specific order. It can also refer to the act of rearranging objects in a specific order.

4. How is algebra used in real life?

Algebra is used in many real-life situations, such as calculating and managing finances, solving engineering problems, and understanding and analyzing data in fields like economics and science.

5. What is the difference between a permutation and a combination in algebra?

A permutation involves arranging a set of objects in a particular order, whereas a combination does not consider the order of the objects. Additionally, in a permutation, each object can only be used once, while in a combination, objects can be repeated.

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