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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:

    [tex]
    H,gH,g^{2}H,...,g^{m-1}H
    [/tex]

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

    [tex]
    gH,g^{2}H,...,g^{m}H
    [/tex]

    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])?
     
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