Answer to Question #300292 in Abstract Algebra for ksmoorthy

Question

F ind aba -1 w here (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) $a = (5 , 7 , 9),\ b = (1 , 2, 3)$

Since $a^{-1} = (5 , 9 , 7),$ it follows from the fact that the circles $a$ and $b$ are independent that

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

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

Since $a^{-1} = (1 , 5,2)(3 , 4),$ it follows that

$aba^{ -1}=(1 , 2,5)(3 , 4) \circ (1 , 4 , 5)\circ (1 , 5,2)(3 , 4) =(1 , 3, 4)(2,5)\circ (1 , 5,2)(3 , 4) \\=(1 , 2, 3)$