Can someone please verify these?
Compute the orbit of a vertex of the square under the tautological action of $D_4$.
Compute the orbit of a point $x \in G$ under the action of $G$ on itself by multiplication.
Compute the orbit of a point $z \in \mathbb{C}$ under the action of the circle group $U(1)$ by multiplication.
Answers:
The orbit of a vertex of the square consists of all of the vertices of the square.
The orbit of a point $x \in G$ consists of all elements of $G$, because for any $g \in G$, $x(x^{-1}g) = g$.
The orbit of a point $z \in \mathbb{C}$ consists of all points on the circle of radius $|z|$.
