# Isomorphic G sets

1. Sep 13, 2009

### daveyinaz

I read in a book on groups and representations that any transitive $$G$$-set is isomorphic to the coset space of some subgroup of $$G$$.
Does this mean we can determine all transitive $$G$$-sets up to isomorphism simply by finding all subgroups of $$G$$?

Just want to make sure that if this is the case that I have in my mind the right idea, so we take $$G = S_3$$, then all transitive $$G$$-sets are up to isomorphism...
$$G / \{e\}, G / \langle (12) \rangle , G / \langle (132) \rangle , G / G$$?

Note I do realize that $$\langle (13) \rangle$$ is also a subgroup of $$S_3$$ but the way I see it the coset space would be the same as $$G / \langle (12) \rangle$$

Last edited: Sep 13, 2009
2. Sep 13, 2009

### Office_Shredder

Staff Emeritus
That's correct, except that the coset space of a subgroup of G is not in general a quotient group (since the subgroup may not be normal)

3. Sep 14, 2009

### daveyinaz

Well that would make sense right? Since a G-set isn't necessarily a group...and if it's isomorphic to some coset space, then that coset space isn't a group either.

4. Sep 14, 2009

### Office_Shredder

Staff Emeritus
Yup, but writing

This is notation for quotient groups. If H is a subgroup of G, the set of cosets is often denoted cos(G)