What does "transitive" mean in group action? By definition, the action of G on A is transitive if there is only one orbit, i.e., given any two numbers a, b ∈ A there is some g ∈ G such that a = g · b.
I want to know why "given any two numbers a, b ∈ A there is some g ∈ G such that a = g · b" is equivalent to "there is only one orbit". Based on what I have learned, the number of the orbits of a is

|O(a)| = |G|/|Ga|

, where Ga is the stabilizer of a.
If there is only one orbit, does it mean that the orbits of all elements of A are the same?
If there is only one orbit, then |O(a)| = 1, so |Ga| = |G|?
I'm so confused with the definition of transitive.
 A: The orbit $O(a)$ of an element $a\in A$ is the set $\{ga:g\in G\}$.
Having one orbit means $O(a)=O(b)$ for any $a,b\in A$, which can be easily proved to be the same as transitivity of the action.
Note that the action can be restricted to any orbit $O(a)$ giving a transitive action.
So actually $A$ is the disjoint union of the orbits.
The size of $O(a)$ is irrelevant in being transitive.
For a simple example consider $A=\{0,1,2,3,a,b\}$ and let $G=\Bbb Z/4\Bbb Z=\{0,1,2,3\}$ act on $A$ by addition modulo $4$ on the numbers and
$$0a=2a=1b=3b=a,\quad
0b=2b=1a=3a=b\,.$$
Here we have two orbits: $\{0,1,2,3\}$ and $\{a,b\}$ of different sizes. Restricting the action of any of them yields a transitive action.
A: That definition of transitive basically says "given $a,b\in A$, $a$ is in the orbit of $b$". Meaning that any two elements of $A$ are always in the same orbit. But since all elements have exactly one orbit, there can only be one orbit in total.
And the other way around: if there is only one orbit, then all elements have the same orbit. So any given element $a$ is in the orbit of any further given element $b$. Which is your definition of transitive.
