Hi,

I know that if G acts on set X then the orbit is

Attachment 11301

But say G=X.

Does it mean that I put two g's into the above?

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- May 6th 2009, 08:52 AMRoland25Orbit of g
Hi,

I know that if G acts on set X then the orbit is

Attachment 11301

But say G=X.

Does it mean that I put two g's into the above? - May 6th 2009, 09:55 AMNonCommAlg
- May 6th 2009, 10:44 AMRoland25
Sorry, I meant G(x) in the above.

What I mean is that if Gx={g*x : g in G}

what is Gg if g is in G.

Would it just be Gg={G} ? - May 6th 2009, 11:04 AMNonCommAlg
it depends how you define the group action $\displaystyle *$. if you define that for all $\displaystyle x,g \in G: \ g*x=gx,$ then $\displaystyle Gx=G.$ but if you define $\displaystyle g*x=gxg^{-1},$ for all $\displaystyle g,x \in G,$ then $\displaystyle Gx=\{gxg^{-1}: \ g \in G \},$

which is the conjugacy class of $\displaystyle x.$ - May 6th 2009, 11:23 AMRoland25
thanks for the help so far.

I think * was multiplication in my case and so Gx=G kinda seems more appropriate.

What I still don't get however is what the actual different between Gx and Gg would be if X=G?

Say X doesn't equal G, would Gx and Gg differ then?

This is where g is in G - May 6th 2009, 11:55 AMNonCommAlg