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Cosets: difference between these two statements

  1. Dec 11, 2013 #1
    Hi all,

    Suppose H is a subgroup of G. For g in G, define fg : G/H > G/H by fg (aH) = gaH for a in G, where G/H is the set of left cosets of H in G.

    What is the difference between these two statements:

    1) for a given aH in G/H, the set {g in G : fg(aH) = aH }

    2) set {g in G : fg = the identity permutation in G/H}

    The identity permutation, in this case, meaning fg(aH) = gaH = aH for all cosets aH

    I know that in part 1, a is given and so we can use a to find the solution set of g, but I struggle to work with part 2 without any concrete information about such an a.
     
  2. jcsd
  3. Dec 12, 2013 #2
    In (2) you demand ##f_g(aH) = aH## for all ##a##. So it is the intersection of sets in (1).
     
  4. Dec 12, 2013 #3
    Oh of course. Wow, I didn't see that. So the solution set could be considered:

    Assuming a_i in G for all i in the index set I,
    {g in G : g lies in the intersection ∏i in I {g = a_i h a_i-1 for some h in H} }

    Not sure if ∏ is the best symbol to represent intersection, but for now let's go with it.

    Thanks for the reply!
     
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