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.
Best Answer
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.