Remark

Recall that the definition of direct sum extends immediately to a family $(E_i)_{i \in I}$ of groups which are not necessarily commutative, multiplicative notation of course replacing the additive notation; we then say "*restricted product*" or "*restricted sum*" instead of "*direct sum*" (I, § 4, no. 9).

Note that $E$ is a normal subgroup of the product $F = \prod_{i \in I} E_i$ and that each of the $j_k(E_k)$ is a normal subgroup of $F$; moreover, for two distinct indices $\lambda, \mu$, every element of $j_\lambda(E_\lambda)$ is permutable with every element of $j_\mu(E_\mu)$. Proposition 6 extends to the general case with the hypothesis that, for two distinct indices $\lambda, \mu$, every element of $f_\lambda(E_\lambda)$ is permutable in $M$ with every element of $f_\mu(E_\mu)$ (I, § 4, no. 9, Proposition 12). The property of "*associativity*" of the restricted sum follows immediately from this. Proposition 7 and its Corollaries 1 and 2 hold without modification.