Theorem 2

Let G be a group with operators. For an equivalence relation R on G to be compatible with the group law and the action of $\Omega$, it is necessary and sufficient that it be of the form $x^{-1}y \in H$, where H is a normal stable subgroup of G (the relation $x^{-1}y \in H$ being moreover equivalent to $yx^{-1} \in H$ for such a subgroup). The law of composition on $G/R$ the quotient of that on G and the action of $\Omega$ on $G/R$ the quotient of that of $\Omega$ on G by such a relation R give $G/R$ the structure of a group with operators, called the quotient structure, and the canonical mapping of the passage to the quotient is a homomorphism of groups with operators.