# 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 $H_i \le G_i$. 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 $a_ib_ia_i^{-1}$.

Since $H_i$ is normal in $G_i$, and $a_i \in G_i, b_i \in H_i$, $a_ib_ia_i^{-1}$ lies in $H_i$. Hence, the $i^{th}$ coordinate of $aba^{-1}$ is in $H_i$ for each $i$, thus $aba^{-1} \in H$.