Normality is direct product-closed
This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Suppose is a nonempty indexing set, and for each , we have a group-subgroup 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 .
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 .