Faithful group actions for singleton group or for empty set

logic elementary-set-theory abstract-algebra group-theory group-actions

Solution 1:

  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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