Corollary

If G is a finite group of order g and H is a subgroup of G of order h, then

(in particular, the order and index of H are divisors of the order of G).

Theorem 1 allows us to determine the equivalence relations compatible with the laws on a group with operators G: if R is such a relation, it is both left and right compatible with the group law on G and with the action of $\Omega$. Hence, if H is the class of $e$ (mod. R), H is a stable subgroup such that the relations $y \in xH$ and $y \in Hx$ are equivalent (since both are equivalent to R); hence $xH = Hx$ for all $x \in G$. Conversely, if this is so, one or other of the equivalent relations $y \in xH, y \in Hx$ is compatible with the group law, since it is both left and right compatible with this law ($\S$ 3, no. 4) and is compatible with the action of $\Omega$. Since the equation $xH = Hx$ is equivalent to $xHx^{-1} = H$, we make the following definition: