When dealing with permutations in permutation notation, "squaring" the permutation means applying the permutation twice in sequence. For a permutation $\sigma$, the square $\sigma^2$ is the result of performing $\sigma$ followed by $\sigma$ again.

For example, if we have a permutation $\sigma$ in cycle notation:

$\sigma = (1 \ 2 \ 3)$

Applying $\sigma$ once means:

• $1$ moves to $2$,
• $2$ moves to $3$,
• $3$ moves to $1$.

Now, squaring $\sigma$ means applying $\sigma$ again, or $\sigma^2$:

• $1$ (which was sent to $2$ by the first application) now moves to $3$,
• $2$ (which was sent to $3$ by the first application) now moves to $1$,
• $3$ (which was sent to $1$ by the first application) now moves to $2$.

So, $\sigma^2 = (1 \ 3 \ 2)$.

In summary:

• Applying a permutation twice results in a new permutation.
• The permutation cycles shift positions according to how the elements are mapped in the first application.

Would you like me to go deeper into cycle notation, or show a detailed example with a different permutation?

Here are 5 related questions to explore further:

1. What is the inverse of a permutation?
2. How do you compose two different permutations?
3. Can a permutation be expressed in multiple forms of cycle notation?
4. What is the order of a permutation?
5. How do you find the identity permutation?

Tip: To compute powers of a permutation, always track each element’s movement step by step.