$H$ is not normal group, take $x \in G$ such that $Hx \ /xH$. That is, there is $h ∈ H$ such that $hx \not \in xH$. Then $hxH \not = H$, and, if the same definition were to work, supposedly $hH ∗ xH = (hx)H \not = xH$ But, on the other hand, since $hH = eH$, $hH ∗ xH = eH ∗ xH = (ex)H = xH$ That is, if H is not normal, this apparent definition is in fact not well-defined. ($H$ is a subgroup of $G$)
What is $Hx \ /xH$?