What does "transitive" mean in group action?

Question

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.

Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. — José Carlos Santos, Nov 04 '21 at 07:41
$|O(a)|$ is the size of the orbit of $a$ not the count of orbits. And yes, transitive means $O(a)=O(b)$ for any $a,b\in A$. — Berci, Nov 04 '21 at 07:48
@Berci so if there is only one orbit, then |O| = |G|, then the stabilizer of all a, b ∈ A is only the identity element? — best learner, Nov 04 '21 at 07:58

score 2 · Answer 1 · answered Nov 04 '21 at 08:56

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.

score 2 · Accepted Answer · answered Nov 04 '21 at 09:03

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.

What does "transitive" mean in group action?

2 Answers2