Normality is direct productclosed
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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Statement
Propertytheoretic statement
The subgroup property of being a normal subgroup is a direct productclosed subgroup property.
Symbolic statement
Suppose is a nonempty indexing set, and for each , we have a groupsubgroup pair . Let be the external direct product of the s, and the subgroup of obtained as the external direct product of the s. Then is a normal subgroup of .
Proof
Using notation from the symbolic statement.
Let . It suffices to show that .
Denote by the coordinates of and . Then the coordinate of is .
Since is normal in , and , lies in . Hence, the coordinate of is in for each , thus .