Find $(a_1, a_2, ..., a_n)^{-1}$

Question

The question that I have to answer is: Find $(a_1, a_2, ..., a_n)^{-1}$. I'm not really sure what this means, or where to start. Our current unit is on permutations, alternating groups, dihedral groups, etc.

Answer 1

Let $\sigma = (a_1 \space a_2 \dots \space a_n)$ to find $\sigma^{-1}$ we wish to find the cycle that when multiplied by $\sigma$ give the identity. (Sending each $a_i$ back to $a_i$)

To do this just brute force reverse the process, so:

$a_1$ needs to get sent back to $a_n$

$a_2$ needs to get set back to $a_1$

In general $a_i$ needs to get sent back to $a_{i-1}$ for $2 \leq i \leq n$

Thus $\sigma^{-1} = (a_n \space a_{n-1} \dots \space a_1)$