Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Isomorphic G sets

  1. Sep 13, 2009 #1
    I read in a book on groups and representations that any transitive [tex]G[/tex]-set is isomorphic to the coset space of some subgroup of [tex]G[/tex].
    Does this mean we can determine all transitive [tex]G[/tex]-sets up to isomorphism simply by finding all subgroups of [tex]G[/tex]?

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

    Note I do realize that [tex]\langle (13) \rangle[/tex] is also a subgroup of [tex]S_3[/tex] but the way I see it the coset space would be the same as [tex]G / \langle (12) \rangle[/tex]
     
    Last edited: Sep 13, 2009
  2. jcsd
  3. Sep 13, 2009 #2

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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)
     
  4. Sep 14, 2009 #3
    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.
     
  5. Sep 14, 2009 #4

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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:H)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook