Question

Find aba -1

where (i) a = (5 , 7 , 9) , b = (1 , 2, 3) (ii) a = (1 , 2,5)(3 , 4) , b = (1 , 4 , 5) .

Answer 1

Let us find $aba^{-1}$ where

(i) $a=(5,7,9),\ b=(1,2,3)$

It follows that $a^{-1}=(5,9,7).$ Taking into account that the circles $a$ and $b$ are independent, and hence they commute, we conclude that

$aba^{-1}=(5,7,9) (1,2,3)(5,9,7)=(1,2,3)(5,7,9)(5,9,7)=(1,2,3)=b.$

(ii) $a=(1,2,5)(3,4),\ b =(1,4,5)$

It follows that $a^{-1}=(3,4)(1,5,2).$ Therefore,

$aba^{-1}=((1,2,5)(3,4))(1,4,5)((3,4)(1,5,2))=((134)(2,5))((3,4)(1,5,2)) =(123).$