Characterizing transitive G-set actions in terms of orbits

• xitoa
X?In summary, for a group G to act transitively on a nonempty G-set S, for any two elements s1 and s2 in S, there exists an element g in G such that g*s1 = s2. This means that every element in S can be reached from any other element through the action of some element in G. In terms of orbits, this means that the orbit of an element x in S is the set of all elements in S that can be reached from x through the action of G. This can be proven by the definition of transitive actions, which states that for each x and y in S, there must exist a g in G such

Homework Statement

A group G acts transitively on a non empty G-set S if, for all s1, s2 in S, there exists an element G in G such that g*s1 = s2. Characterize transitive G-set actions in terms of orbits. Prove your answer

Homework Equations

Transitive G-set Actions: for all s1, s2 in S, there exists a g in G such that g*s1=s2
Regular G-set Actions: 1) (gh)s = g(hs)
2) 1s = s

Orbit of S = {s' in S such that s' in gs for some g in G}

The Attempt at a Solution

Past the definitions, i don't really know anything. The problem is vague and it seems like I'm supposed to write the definition of transitive G-set actions in terms of orbits, but i don't know how to do that nor do i know how i would "prove" that.

Tips are greatly appreciated :D.

The orbit of an element x is the set

$$\{g\cdot x~\vert~g\in G\}$$

So the orbit is all the elements where you could send x to. What possible element can you send x to in a transitive action?? What does transitive mean??

transitive: a(bc) = (ab)c

and could i send x to itself transitively? say a(a^-1x) = (aa^-1)x?

xitoa said:
transitive: a(bc) = (ab)c

and could i send x to itself transitively? say a(a^-1x) = (aa^-1)x?

No, that's not what transitive is.

oh shoot. yes that's totally wrong lol...

if a = b, b = c, then a = c.

edit: could i be sending x to the whole set of X? could i send it to the whole thing or only one element of X? I'm not sure.

Last edited:
No...

Search in your notes for "transitive action". What is the definition they give. Don't just make things up...

the only definition i have for transitive action is transitive action on groups and I've posted it above...ahh:/

xitoa said:
the only definition i have for transitive action is transitive action on groups and I've posted it above...ahh:/

You already posted it in the OP... An action is transitive if for each x and y there is a g such that $g\cdot x=y$. I don't know where the other things come from.

Now what does transitive mean intuitively?? Can you calculate the orbit right now??

oh well i thought you were asking for something other than the OP o.o

intuitively...the transitive action takes one orbit to another orbit?

the orbit should be {g*x} for x in X

No, the action always sends an element to the same orbit. By definition.

Transitivity says that every number can be sent to every other number.

Now, with this, what is the orbit??

if an action is transitive, how can x,y lie in different orbits?