I want to work out the order of
$\alpha = \left[ \begin{align} & 1&2&&3&&4&\\&4&2&&1&&3& \end{align} \right]$ in $S_4$
Now when I think of $S_4$ I think of a piece of paper that looks like this:
$S_4 = \left[ \begin{align} &1&2\\&4&3 \end{align} \right]$
So $\alpha = \left[ \begin{align} &4&2\\&3&1 \end{align} \right]$
How can I work out the order of this?
I am fairly sure I have three operations being:
$Op_1 = \left[ \begin{align} & 1&2&&3&&4&\\&4&1&&2&&3& \end{align} \right]$
$Op_2 = \left[ \begin{align} & 1&2&&3&&4&\\&4&3&&2&&1& \end{align} \right]$
$Op_3 = \left[ \begin{align} & 1&2&&3&&4&\\&2&1&&4&&3& \end{align} \right]$
Is this correct?
Also is there a way to disable the $LaTeX$ autorendering below my editing box? It is freezing my old laptop to a stop.
The order of $\alpha$ is the smallest positive integer $n$ such that $\alpha^n$ is the identity permutation.
Trace what happens to $1$. It is taken to $4$ which is taken to $3$ which is taken to $1$. In symbols, $\alpha^3$ applied to $1$ is $1$.
Note that $2$ is taken to $2$. Also, $\alpha$ takes $3$ to $1$ which is taken to $4$ which is taken to $3$. The history of $4$ is similar. Thus $3$ is the smallest positive integer $n$ such that $\alpha^n$ is the identity.
Remark: Somewhat more generally, let $\pi$ be a permutation of a finite set $\{a_1,\dots,a_n\}$. For each $i$, let $n_i$ be the smallest positive integer such that $\pi^{n_i}(a_i)=a_i$. Then the order of $\pi$ is the least common multiple of the $n_i$.