The Power of Permutations

The power of a permutation refers to applying a permutation $ \sigma $ to itself $ k $ times, denoted as $$ \sigma^k $$.

A permutation is essentially a rearrangement of a set's elements.

When we take a permutation $\sigma$ of a finite set and apply it to itself $k$ times, we get what's called the $k$th power of $\sigma$ (represented as $\sigma^k$).

The order of a permutation $\sigma$ is defined as the smallest positive integer $n$ such that $\sigma^n$ is the identity permutation, which leaves every element in its original position.

Example

Let's consider a permutation $\sigma$ on the set $ X = \{1, 2, 3\} $ defined in cycle notation as $\sigma = (1\ 2\ 3)$.

Applying $\sigma = (1\ 2\ 3)$ once, we observe:

1 is moved to 2,
2 is moved to 3,
3 is moved back to 1.

Alternatively, this permutation can be depicted in matrix form as $$ \sigma = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{pmatrix} $$

Let's explore the initial powers of this permutation:

$\sigma^1 = \sigma$ simply cycles 1 to 2, 2 to 3, and 3 back to 1.
$\sigma^2 = (1\ 3\ 2)$ takes it a step further. By reapplying $\sigma$, each element progresses one more position along the cycle.
To illustrate, since 1 was initially moved to 2, it now progresses to 3 upon the second application of $\sigma$. Similarly, 2, which moved to 3, now cycles back to 1. Repeating this pattern gives us the configuration $\sigma^2$.
$\sigma^3 = id$ brings every element back to its starting point, completing the cycle and restoring the original configuration. This results in the identity permutation.
Here’s how it happens: 1 goes to 2, then 3, and finally back to 1. Each element follows this path, which is why, after three applications, all elements return to their original positions.

In this example, the order of $\sigma$ is 3, as it takes three applications to return to the identity permutation.

Permutations can be analyzed in terms of cycles, groups of elements that cyclically shift among themselves. Understanding these cycles can elucidate the effects of raising a permutation to successive powers.