 Faithful quotient of a group action

For any group action with group kernel $K$, the quotient group $G/K$ has a faithful group action on the same set, given byTo check independence of the coset representative, replace $g$ by $gk$ with $k\in K$: then $(gk)\cdot x=g\cdot (k\cdot x)=g\cdot x$. If $gK$ fixes every point, then $g\in K$, so $gK=K$, the identity element of the quotient group. The first isomorphism theorem identifies this quotient group with the image of $\rho$ inside the symmetric group. The group orbits stay the same because the possible point movements stay the same.

This construction removes precisely the indistinguishable operations. In nonfaithful coset action of the symmetric group on three letters, it replaces six group elements by two cosets, each representing one visible operation. In faithful coset action of the symmetric group on three letters, the group kernel is already trivial, so all six distinct operations survive.