Remark

The construction is a generalisation of the left regular representation of G. The latter is the special case where H = {e}, the trivial subgroup of G.

If gH is a left coset of H, then the stabilizer of gH is the conjugate of H by g, i.e. the stabilizer equals gHg^-1 = {ghg^-1 | h H}.

Indeed, for each element ghg^-1, with h H, we have ghg^-1gH = ghH = gH. And, on the other hand, if k G satisfies kgH = gH, then there is a h H with kg = gh, from which we deduce k = ghg^-1.