Title is the question. I'm trying to understand these definitions better.
Is the orbit of an element simply the possible values of the element when a group action is applied to it?
If so, why the abstract definition with equivalence relations and similar? It would seem like a simple concept.
The definition of action, which is property-based and not given by a closed formula, is patterned upon the basic properties fulfilled by the "prototypical" action, namely the natural acting (literally) of the bijections on a set $X$ on the elements $x \in X$: \begin{alignat}{1} &\iota_X(x)=x, \forall x \in X \\ &(\sigma\tau)(x)=\sigma(\tau(x)), \forall \sigma,\tau \in \operatorname{Sym}(X), \forall x\in X \end{alignat} The generalization to abstract groups works because in turn the definition of abstract group is precisely patterned upon the properties fulfilled by the set of bijections on a set $X$, endowed with the composition as operation (closure, associativity, identity, inverses). In particular, the "prototypical" version of orbit's definition in abstract group context, namely $O(x)=\{g\cdot x, \space g\in G\}$, becomes in the "native" context of bijection on a set $X$, $O(x)=\{\sigma(x), \space \sigma\in \operatorname{Sym}(X)\}$.