I have two statements about group actions and have to determine which of them are true and why.
The question goes as follows:
Let $X$ be a set and $G$ a group with $g \in G$. Let $\alpha_g $ be an action of the group $G$ in the set $X$. Determine which of the following is true.
I) If $\alpha_g(x_1) = \alpha_g(x_2)$, then $x_1 = x_2$.
II) If $\alpha_{g_1}(x) = \alpha_{g_2}(x)$, then $g_1 = g_2$.
I've got a hunch that statement II may be false, but cannot determine why.
I verified the core properties of group actions and searched for some examples but couldn't find none that matched any of these statements.
Would anyone be able to help?
For I, recall that a group action is a homomorphism $G \to \operatorname{Aut} S$ where $\operatorname{Aut} S$ denotes the group of bijections from $S$ to itself, given by $g \mapsto \alpha_g$. Hence $\alpha_g$ is injective. (If this is not your definition of a group action, it is a good exercise to show that $\alpha_g$ is a bijection.)
For II, consider the trivial action $g \mapsto \text{Id}_S$.