Find All Transitive G-Sets Up to Isomorphism w/ Subgroups of G

[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 $$G / \langle (12) \rangle$$

 

[Office_Shredder]

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)

 

[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.

 

[Office_Shredder]

Yup, but writing

$$G / \{e\}, G / \langle (12) \rangle , G / \langle (132) \rangle , G / G$$

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