How do i find the order of a permutation in the group $S_n$

Question

How can i define the order of a permutation without doing the permutation again and again? Example: say $σ=(1-->2,2-->3,3-->5,4-->1,5-->4)$ in $S_5$.

Answer 1

Write the permutation in cycle notation (and expressed as the product of disjoint cycles).

For each cycle in the permutation determine its length. (If there is only one cycle of length greater than $1$, then the order of the permutation is the length of that cycle. If all cycles in a permutation are of length one, it is necessarily the identity permutation, and has order $1$.

Calculate the least common multiple of all cycle lengths.

The result is the order of the permutation.

Example: Let $\tau =\begin{pmatrix} 1& 2& 3& 4& 5\\ 2& 4 & 5 & 1& 3\end{pmatrix}$

Written as the product of disjoint cycles gives us $\tau = (1, 2, 4)(3, 5) \in S_5$.

Then the lengths of the cycles, from left to right, are $3, 2$.

$\operatorname{lcm}(3, 2) = 6$.

The order of $\tau$ is equal to $6$.

Answer 2

The order is the least common multiplier of the length of all circles in the circle notation of the permutation.

Answer 3

The element that you have listed isn't permutation because it sends both $4$ and $5$ to $1$. If, however, you had $\sigma = (1235) \in S_5$, then you can find the order by simply multiplying the permutation by itself: $$\begin{align} \sigma^1 &= (1235) \\ \sigma^2 &= (13)(25) \\ \sigma^3 &= (1532) \\ \sigma^4 &= (1). \end{align} $$ So the order of $\sigma$ is $4$.

This, obviously, only works when the group is relatively small.

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If you wanted to ask about the order of $\sigma = (12354)$, then just star $$\begin{align} \sigma^1 &= (12354) \\ \sigma^2 &= (13425) \\ \sigma^3 &= \dots. \end{align} $$

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In case you are not familiar with the notation, $(1235)$ is the permutation that $$\begin{align} 1 &\longrightarrow 2 \\ 2 &\longrightarrow 3 \\ 3 &\longrightarrow 5 \\ 5 &\longrightarrow 1\\ 4 &\longrightarrow 4. \end{align} $$