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 $\mu(x,g) = x$.


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