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

Question about Lagranges theorem in Group Theory

  1. Jan 3, 2010 #1
    1. The problem statement, all variables and given/known data
    If H is a subgroup of a finite group G, and if the order of G is m times the order of H, |G|=m|H|, adapt the proof of Lagrange's theorem to show that gm! is an element of H for all g in G.

    3. The attempt at a solution
    My thoughts so far were to think that we can divide G into m pieces: H, g1H, g2H, ..., gm-1H. If we now take an arbitrary gk, it is certainly in gkH, since H contains the identity. The question now is: what happens if we look at gk2, (gk2)3... where do they end up? I need a hint...
  2. jcsd
  3. Jan 3, 2010 #2
    If g is in H, there's nothing to prove; if not, consider the m (left) cosets:


    Now, if you apply the translation by g again, you get:


    But this translation is bijective, and that implies that [tex]g^{m}H[/tex] must be equal to what? And from this, what may be inferred about [tex]g^{m}[/tex] (and also [tex]g^{m!}[/tex])?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook