Finding $(ab)^{-2}$ and $(a^{-1})(b^{-1})$ for $a=(236)(57), b=(147).$

Question

I have a problem with following example:

$a = (236)(57), b = (147)$

I need to find

1. $(ab)^{-2}$
2. $(a^{-1})(b^{-1})$

How to solve it? I know, how to find $(ab)^{-1}$:

$$((236)(1475))^{-1} = (263)(1574)$$

But how can I multiply $(263)(1574)(263)(1574)$ in order to get $(ab)^{-2}$?

Answer 1

Since the basic cycle formations are disjoint you have $$ (263)(1574)(263)(1574) = (263)(263)(1574)(1574) $$ which amounts to shifting around the same cycle twice, i.e. $(1574)^2 = (17)(54)$.

Answer 2

First do $$(263)(1574)(263)(1574)=(263)^2(1574)^2=(236)(17)(45)$$

Then the inverse is $$(263)(17)(45)$$