This has been brought up here but I'd like to bring up a few more questions.
Taking the quotient of group $G$ with subgroup $H$ is well-defined iff $H$ is normal in $G$. Well, what happens when $H$ is not normal? The left- and right cosets of $H$ in $G$ don't coincide. I have this situation in a paper I'm reading; quotients are taken with not necessarily normal subgroups. Despite that, this never seems to be an issue with the authors; no "note that this quotient is not well-defined" or special theorems for "non-normal quotients".
Here's an example: Let $H \subset L \subset S_n$ where $H=Stab_L(F)^*$ and $H' := Stab_{S_n}(F)$. Then $[S_n : L][L:H] = [S_n : H'][H':H]$.
These are indices of left cosets. Does the above equality make sense even if the associated quotient groups are not well-defined?
(*I don't think it's relevant here what $F$ actually is, but see my other questions and you can probably guess.)
Best Answer
The index of a subgroup is always well-defined. The number of left cosets of a subgroup is always the same as the number of right cosets. Normality only comes in when you want the cosets to acquire a natural group structure.