Let $G$ be a finite group and $H$ a subgroup. Is it true that
a set of right coset representatives of $H$ is also set of left coset
representatives of $H$?
[Math] Relation between left and right coset representatives of a subgroup
group-theory
group-theory
Let $G$ be a finite group and $H$ a subgroup. Is it true that
a set of right coset representatives of $H$ is also set of left coset
representatives of $H$?
Best Answer
Not every left transversal is also a right transversal. The Group Properties Wiki has a list of subgroup properties that are stronger than "having [at least one] left transversal that is also a right transversal"; among these is normality as William notes.
However the left coset representatives' multiplicative inverses form a right transversal, because
$$\begin{array}{c l}xH=yH & \iff y^{-1}xH=H \\ & \iff y^{-1}x\in H \\ & \iff (y^{-1}x)^{-1}=x^{-1}y\in H \\ & \iff Hx^{-1}y=H \\ & \iff Hx^{-1}=Hy^{-1}. \end{array}$$