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

[decerto]

Homework Statement

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

Homework Equations

$Parity(\sigma)= (\sum_{i=1}^k(|c_i|-1)) mod 2$

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.

 

[Dick]

Homework Statement

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

Homework Equations

$Parity(\sigma)= (\sum_{i=1}^k(|c_i|-1)) mod 2$

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 $\sigma^5$?

 

[kduna]

What does σ⁵ being even tell you?