# Faithful group actions for singleton group or for empty set

Let $$G$$ be a group, possibly a singleton. Let $$M$$ be a set, possibly empty. Let $$\mu: M \times G \to M$$ be a right group action. Let $$1_G$$ be the identity of $$G$$.

I understand Wikipedia's second definition of faithful as follows:

$$\mu$$ is faithful if for each $$g \in G$$ such that $$g \ne 1_G$$, there exists $$x \in M$$ such that $$\mu(x,g) \ne x$$. $$\tag{1}$$

Later Wikipedia says if $$\mu$$ is free and $$M$$ is a non-empty set, then $$\mu$$ is faithful.

Question: Given the idea that we can have $$M$$ as an empty set, I would like to clarify: Are these correct?

1. If $$G$$ is a singleton, then every action $$\mu$$ is faithful, whether or not $$M$$ is empty.

2. If $$G$$ is not a singleton, but $$M$$ is empty, then every action $$\mu$$ is not faithful.

1. Correct. If $$G=\{1\}$$, then every action is faithful since the condition holds vacuously.
2. Also correct. If $$G\neq \{1\}$$, then there exists $$g \in G$$ such that $$g \neq 1$$. However $$M$$ is empty, so there does not exist $$x \in M$$ such that $$\mu(x,g) = x$$.