Inverse permutation

Question

Can anyone help me with this question. ((14638)(259))^-1

I understand the normal way of multiplying permutation with cycle notation but here is an inverse sign so I don’t know what to do?

Many thanks.

Answer 1

Hint: The cycles are disjoint and so the inverse of $\pi\rho= (14638)(259)$ is $(\pi\rho)^{-1} = \rho^{-1}\pi^{-1}$. So you have to form the inverse of cycles, which is quite easy:

The inverse permutation of $(x_1x_2\ldots x_n)$ is $(x_1x_n\ldots x_2)$.

Answer 2

In a group, $(xy)^{-1}=y^{-1}x^{-1}.$ And the inverse of a cycle can be obtained by reversing the order of entries, e.g. $(acbd)^{-1}=(dbca).$