Is there a relationship between right cosets and orbits in group theory?

[cmj1988]

I'm just wondering if there is some sort of relationship between right coset and orbit of x. We just got to cosets, and it seems like the properties of cosets are eerily similar to orbits.

 

[whs]

Of course their is a relationship.

First of all, I think you answered your own question.

Second, a random book I picked up had a section on cosets right after the section on orbits and cyclic groups.

Q.E.D.

 

[whs]

I'd like to note, you specifically mentioned the right coset and orbit of 'x'. Be careful if you choose a different variable, say, 'z'.

Z has mystical properties that defy abstract algebra.

 

[Fredrik]

I'm just wondering if there is some sort of relationship between right coset and orbit of x. We just got to cosets, and it seems like the properties of cosets are eerily similar to orbits.

You might be interested in theorem 3 on page 4 of this article.

The study of group actions thus divides neatly into two problems: the internal problem of understanding the action within single orbits (equivalent to studying the canonical action in coset spaces) and the external problem of understanding how the orbits are put together to form the set X.