Simple Algebra Proof: Finding all elements of a Group, S.

[ThatPinkSock]

Homework Statement

Find all elements x in S₄ such that x⁴ = e.

(e is the identity)

Homework Equations

Cycle notation is advised.

The Attempt at a Solution

I know there are 4 elements in this (1 2 3 4). And I feel like I need to find the order of S which can be found by finding the number of cycles.

Suppose S is a finite group with x elements, then xⁿ = e for all x in S. That's my start. not sure where to go from there.

 

[Adrmay]

A:Let's start by understanding what the question is asking. The group $S_4$ is the symmetric group on 4 elements (the group of all bijections from a set of $4$ elements to itself). An element $x\in S_4$ is an element of the group which is a bijection from the set of 4 elements to itself. We want to find all $x\in S_4$ such that $x^4=e$, where $e$ is the identity element of the group.The first thing to note is that $S_4$ is a finite group, so it has order $4!=24$. This means that for any $x\in S_4$, the order of $x$ divides $24$. Since $x^4=e$ implies $x^{24}=e$, the order of $x$ must divide both $4$ and $24$. The only divisors of both $4$ and $24$ are $1$, $2$, and $4$, so the order of $x$ must be one of these numbers.Now, let's consider each possibility in turn. If the order of $x$ is $1$, then $x^1=x=e$, so $x=e$ is the only possibility. If the order of $x$ is $2$, then $x^2=