Diagonal subgroup of a direct power

Definition
Suppose $$G$$ is a group and $$H = G^S$$ is an unrestricted external direct product of $$|S|$$ copies of $$G$$. Equivalently, $$H$$ can be thought of as the group of functions from $$S$$ to $$G$$ with the group operations performed pointwise. Then, the diagonal subgroup is the subgroup comprising the constant functions from $$S$$ to $$G$$, or equivalently, the elements of $$H$$ with all coordinates equal.

Weaker properties

 * Stronger than::Base diagonal of a wreath product