Proposition 15

Let G be a group with operators and $(H_i)_{1 \leq i \leq n}$ a sequence of normal stable subgroups of G such that

the set $H_1 H_2 \ldots H_n$ is a normal stable subgroup of G, the restricted sum of the $H_i$.

By induction on $n$, this is immediately reduced to proving the proposition for $n = 2$. We show first that, if $x \in H_1$ and $y \in H_2$, x and y are permutable; for $xyx^{-1}y^{-1} = (xyx^{-1})y^{-1} = x(yx^{-1}y^{-1})$ and hence (since $H_1$ and $H_2$ are normal) $xyx^{-1}y^{-1} \in H_1 \cap H_2$, that is $xyx^{-1}y^{-1} = e$, by the hypothesis. Moreover $H_1 H_2$ is a subset of G which is stable under the homotheties of G. It follows (by no. 3, Proposition 1) that $H_1 H_2$ is a stable subgroup of G and it is immediately verified that this subgroup is normal. Suppose finally that $xy = x'y'$, with $x \in H_1, x' \in H_1, y \in H_2, y' \in H_2$; then ${x'}^{-1}x = y'y^{-1}$, hence ${x'}^{-1} \in H_1 \cap H_2 = \{e\}$, $x' = x$ and similarly $y' = y$; $H_1 H_2$ is thus the direct product of $H_1$ and $H_2$.

When the group considered are commutative, the term direct sum is used instead of direct product.