Group orbit computations

Question

Can someone please verify these?

1. Compute the orbit of a vertex of the square under the tautological action of $D_4$.

2. Compute the orbit of a point $x \in G$ under the action of $G$ on itself by multiplication.

3. Compute the orbit of a point $z \in \mathbb{C}$ under the action of the circle group $U(1)$ by multiplication.

Answers:

1. The orbit of a vertex of the square consists of all of the vertices of the square.

2. The orbit of a point $x \in G$ consists of all elements of $G$, because for any $g \in G$, $x(x^{-1}g) = g$.

3. The orbit of a point $z \in \mathbb{C}$ consists of all points on the circle of radius $|z|$.

Answer 1

Your answers are correct. ${}{}{}{}{}$