TITLE: Does this group action construction have a name?
QUESTION [7 upvotes]: Let $G \curvearrowright X$ be a group action. Then $G \curvearrowright X \times X$ through $g \cdot (x, y) = (g \cdot x, g \cdot y)$. I am interested if this 'diagonal' action and its orbits have a special name.

REPLY [4 votes]: If I remember correctly, your guess "diagonal action" is a common name for this kind of group action.