Relation between direct product of subgroups and direct product of groups

For two groups
Suppose $$G_1$$ and $$G_2$$ are groups. Suppose $$H_1$$ is a subset of $$G_1$$ and $$H_2$$ is a subset of $$G_2$$. Then, we can consider $$H_1 \times H_2$$ as the subset of the external direct product $$G_1 \times G_2$$ where the first coordinate is in $$H_1$$ and the second coordinate is in $$H_2$$.

We then have the following:


 * 1) $$H_1 \times H_2$$ is a subgroup of $$G_1 \times G_2$$ if and only if $$H_1$$ is a subgroup of $$G_1$$ and $$H_2$$ is a subgroup of $$G_2$$. Further, in this case, the group structure on $$H_1 \times H_2$$ viewing it as a subgroup coincides with its group structure viewed abstractly as an external direct product.
 * 2) $$H_1 \times H_2$$ is a normal subgroup of $$G_1 \times G_2$$ if and only if $$H_1$$ is a normal subgroup of $$G_1$$ and $$H_2$$ is a normal subgroup of $$G_2$$. Further, in this case, the quotient group $$(G_1 \times G_2)/(H_1 \times H_2)$$ can be identified naturally with the external direct product of the individual quotient groups, i.e., the external direct product $$G_1/H_1 \times G_2/H_2$$.

For finitely many groups
Suppose $$G_1,G_2,\dots,G_n$$ are groups. For each $$i$$, consider a subset $$H_i$$ of $$G_i$$. Consider $$H_1 \times H_2 \times \dots \times H_n$$ as the subset of the external direct product $$G_1 \times G_2 \times \dots \times G_N$$ where the $$i^{th}$$ coordinate is in $$H_i$$ for all $$i \in \{ 1,2,\dots,n\}$$.

We then have the following:


 * 1) $$H_1 \times H_2 \times \dots \times H_n$$ is a subgroup of $$G_1 \times G_2 \times \dots G_n$$ if and only if each $$H_i$$ is a subgroup of the corresponding $$G_i$$. Further, in this case, the group structure on $$H_1 \times H_2 \times \dots \times H_n$$ viewing it as a subgroup coincides with its group structure viewed abstractly as an external direct product.
 * 2) $$H_1 \times H_2 \times \dots \times H_n$$ is a normal subgroup of $$G_1 \times G_2 \times \dots G_n$$ if and only if each $$H_i$$ is a normal subgroup of the corresponding $$G_i$$. Further, in this case, the quotient group $$(G_1 \times G_2 \times \dots G_n)/(H_1 \times H_2 \times \dots \times H_n)$$ can be identified naturally with the external direct product of the individual quotient groups, i.e., the external direct product $$G_1/H_1 \times G_2/H_2 \times \dots \times G_n/H_n$$.