I have two problems involving equivalence relations. I will only ask the first for now, in the hopes of figuring out the second one after this one is clear.

Let $X$ be a set, and $G$ be a subgroup $G \subset S\_X$. Consider the relation on $X$ given by $x \equiv\_G y$ if there's an element $g \in G$ s.t. $g(x)=y$. Prove that this is an equivalence relation on $X$.

Attempt: Let $E$ be an equivalence relation on $X$. Then, we have:

1. $(x,y)\in E \Rightarrow (y,y)\in E \Rightarrow (g(x),g(x))\in E$ since $g$ is bijective. So $G$ is reflexive on $X$.
2. Assume $(g(x),g(y))\in E \Rightarrow (y,x)\in E \Rightarrow (x,y)\in E \Rightarrow (g(y),g(x))\in E$. So $G$ is symmetric.
3. To prove transitivity, I think I need a second fact such as $y \equiv\_G z$. I am not sure.

I am convinced that I have erred in notation at some point. The notation I see varies greatly in this area of equivalence.