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I am trying to understand/solve the following...Find the disjoint cycle notation of $(a_1, a_2, . . . , a_n)^{−1}$.

I believe $(a_1, a_2, . . . , a_n)^{−1}= (a_1, a_n, a_{n-1}, ..., a_2), $ but is this in disjoint cycle notation? It's probably just because it's written with a variable $a$ instead of being given a finite number, but I'm a little confused because I know if you have a permutation like $\begin{pmatrix} 1 & 2 &3 &4 &5 &6 &7 &8 \\ 7& 8 & 4 & 3 &6 &5 &2 & 1 \end{pmatrix} $ then the disjoint cycle notation would be $(1,7,2,8)(3,4)(5,6)$.

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The key point is that disjoint cycles commute and so if $\sigma_1, \dots, \sigma_k$ are disjoint cycles, then $(\sigma_1 \cdots \sigma_k)^{-1}=\sigma_1^{-1} \cdots \sigma_k^{-1}$. As you have observed, $\sigma_j^{-1}$ is easy to find.

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