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

<br /> G / \{e\}, G / \langle (12) \rangle , G / \langle (132) \rangle , G / G<br />

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