Normality is direct product-closed

This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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Statement

Property-theoretic statement

The subgroup property of being a normal subgroup is a direct product-closed subgroup property.

Symbolic statement

Suppose $I$ is a nonempty indexing set, and for each $i\in I$, we have a group-subgroup pair $H_{i}\leq G_{i}$. Let $G$ be the external direct product of the $G_{i}$s, and $H$ the subgroup of $G$ obtained as the external direct product of the $H_{i}$s. Then $H$ is a normal subgroup of $G$.

Proof

Using notation from the symbolic statement.

Let $a\in G,b\in H$. It suffices to show that $aba^{-1}\in H$.

Denote by $a_{i},b_{i}$ the $G_{i}$-coordinates of $a$ and $b$. Then the $G_{i}$-coordinate of $aba^{-1}$ is $a_{i}b_{i}a_{i}^{-1}$.

Since $H_{i}$ is normal in $G_{i}$, and $a_{i}\in G_{i},b_{i}\in H_{i}$, $a_{i}b_{i}a_{i}^{-1}$ lies in $H_{i}$. Hence, the $i^{th}$ coordinate of $aba^{-1}$ is in $H_{i}$ for each $i$, thus $aba^{-1}\in H$.