Definition 5

Let G be a group with operators. A stable subgroup H of G is called a normal (or invariant) stable subgroup of G if $xHx^{-1} = H$ for all $x \in G$.

If $\Omega = \varnothing$, a normal stable subgroup of G is called a normal (or invariant) subgroup of G. In a commutative group every subgroup is normal.

To verify that a stable subgroup H is normal, it suffices to show that $xHx^{-1} \subset H$ for all $x \in G$; for if so then $x^{-1}Hx \subset H$ for all $x \in G$, that is $H \subset xHx^{-1}$, and hence $H = xHx^{-1}$.

Let H be a normal stable subgroup of G and R the equivalence relation $y \in xH$ defined by H; on the quotient set $G/R$, the internal law, the quotient by R of the law of the group G, is associative; the class of e is the identity element of this quotient law; the classes of two inverse elements in G are inverses under the quotient law and the action of $\Omega$, the quotient by R of the action of $\Omega$ on G, is distributive with respect to the internal law on $G/R$ ($\S$ 3, no. 5). Hence, summarizing the results obtained: