I'm trying to prove that given a group $G$ and a subgroup $H$ where $H\leq G$ , and given that $aH = bH$ , then we need to check if also $Ha = Hb$ .
I'm trying to show that this is wrong with a counter example :
$G=S_{3}$ , $H={(1),(1 2)}$ , $a=(1 3) , b=(1 2 3)$
And now we check : $aH = (1 3)H=(1 3) ( (1),(1 2))={(1 3) ,(1 3)(1 2)} = ? $
Here , for the multiplication $(1 3)(1)$ : 1 goes to 1 and 1 goes to 3 , then the result is 1 goes to 3 .
For the second multiplication $(1 3)(1 2)$ : 1 goes to 2 , and then 1 goes to 3 ? how do I calculate this one ?
The same for bH :
$bH=(1 2 3)H=(1 2 3) ( (1),(1 2))={(1 2 3) ,(1 2 3)(1 2)} = ? $
How do I calculate the multiplication of $(123)(12)$ ?
Regards