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Normal subgroups

  1. Mar 2, 2008 #1
    [SOLVED] normal subgroups

    1. The problem statement, all variables and given/known data
    My book states the following without any justification right before proving the Third Isomorphism Theorem: "If H and K are two normal subgroups of G and [itex]K \leq H[/itex], then H/K is a normal subgroup of G/K."
    The elements of H/K are cosets of K in H. The elements of G/K are cosets of K in G. Therefore I think that statement is simply absurd. That is, the elements of H/K are not even contained in the quotient group G/K; therefore, they cannot possibly form a normal subgroup in G/K.

    EDIT: wait, never mind, the cosets of K in H are also cosets of K in G; sorry
    EDIT: and the reason H/K is normal in G/K is that gK*hK*g^(-1)K = (ghg^{-1})K = h' K since H is normal in G. Very EDIT: cool.

    2. Relevant equations

    3. The attempt at a solution
    Last edited: Mar 2, 2008
  2. jcsd
  3. Mar 2, 2008 #2
    The elements in H/K are, indeed, contained in G/K since H is contained in G: A general element in H/K is hK which is in G/K since h is in G.
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