Normality is direct product-closed

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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_is, and H the subgroup of G obtained as the external direct product of the H_is. 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.