# Characterizing transitive G-set actions in terms of orbits

## 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.

micromass
Staff Emeritus
Homework Helper
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?

micromass
Staff Emeritus
Homework Helper
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 thats 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:
micromass
Staff Emeritus
Homework Helper
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:/

micromass
Staff Emeritus
Homework Helper
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

micromass
Staff Emeritus