Show that the center of a direct product is the direct product of the centers: . Deduce that a direct product of groups is abelian if and only if each of the factors is abelian.
We proceed by induction on .
For the base case , let and be groups. Let . Then for all , . Thus and for all and ; hence and . Thus . Let . Now for all and , . Thus . Thus .
Now suppose the conclusion holds for any product of groups for some . Then . By induction, the result holds for any finite direct product.
Now let be a finite direct product of groups.
If each factor is abelian, then ; hence is abelian.
If is abelian, then . If we let denote the th coordinate projection, we have . Hence is abelian for each .