Theorem 1

Let R be an equivalence relation on a group with operators G; if R is left (resp. right) compatible (§ 3, no. 3) with the group law on G and compatible with the action of $\Omega$, the equivalence class of e is a stable subgroup H of G and the relation R is equivalent to $x^{-1}y \in H$ (resp. $yx^{-1} \in H$). Conversely, if H is a stable subgroup of G, the relation $x^{-1}y \in H$ (resp. $yx^{-1} \in H$) is an equivalence relation which is left (resp. right) compatible with the group law on G and compatible with the action of $\Omega$ and under which H is the equivalence class of e.

We restrict our attention to the case where the relation R is left compatible with the law on G (the case of a right compatible relation follows by replacing the law on G by the opposite law). The relation $y \equiv x \pmod{R}$ is equivalent to $x^{-1}y \equiv e \pmod{R}$, for $y \equiv x$ implies $x^{-1}y \equiv x^{-1}x = e$ and conversely $x^{-1}y \equiv e$ implies $y = x(x^{-1}y) \equiv x$. If H denotes the equivalence class of e, the relation R is then equivalent to $x^{-1}y \in H$. We show that H is a stable subgroup of G. For every operator $\alpha$, the relation $x \equiv e$ implies $x^\alpha \equiv e^\alpha = e$, hence $H^\alpha \subset H$ and H is stable under the action of $\Omega$. It suffices to establish (Proposition 1) that $x \in H$ and $y \in H$ imply $x^{-1}y \in H$, that is $x \equiv e$ and $y \equiv e$ imply $x \equiv y$, which is a consequence of the transitivity of R.

Conversely, let H be a stable subgroup of G; the relation $x^{-1}y \in H$ is reflexive since $x^{-1}x = e \in H$; it is symmetric since $x^{-1}y \in H$ implies $y^{-1}x = (x^{-1}y)^{-1} \in H$; it is transitive, for $x^{-1}y \in H$ and $y^{-1}z \in H$ imply $x^{-1}z = (x^{-1}y)(y^{-1}z) \in H$; it is left compatible with the law of composition on G, for $x^{-1}y = (zx)^{-1}(zy)$ for all $z \in G$; finally, for every operator $\alpha$, the relation $y \in xH$ implies $y^\alpha \in x^\alpha H^\alpha \subset x^\alpha H$ and hence the equivalence relation $x^{-1}y \in H$ is compatible with the action of $\Omega$ on G.

Let G be a group and H a subgroup of G; the relation $x^{-1}y \in H$ (resp. $yx^{-1} \in H$) is also written in the equivalent form $y \in xH$ (resp. $y \in Hx$). Every subgroup H of G thus defines two equivalent relations on G, namely $y \in xH$ and $y \in Hx$: the equivalent classes under these relations are respectively the sets $xH$, which are called left cosets of H (or modulo H), and the sets $Hx$, which are called right cosets of H (or modulo H). By saturating a subset $A \subset G$ with respect to these relations (Set Theory, II, § 6, no. 4), we obtain respectively the sets AH and HA. The mapping $x \mapsto x^{-1}$ transforms left cosets modulo H into right cosets modulo H and conversely.

The cardinal of the set of left cosets (mod. H) is called the index of the subgroup H with respect to G and is denoted by $(G:H)$; it is also equal to the cardinal of the set of right cosets.

If a subgroup K of G contains H, it is a union of left (or right) cosets of H. Since a left coset of K is obtained from K by left translation, the set of left cosets of H contained in a left coset of K has cardinal independent of the latter. Hence (Set Theory, III, § 5, no. 8, Proposition 9):