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Cartesian Product of Permutations?

  1. Nov 13, 2013 #1
    Suppose I was asked if [itex] G \cong H \times G/H [/itex]. At first I considered a familiar group, [itex]G = S_3 [/itex] with its subgroup [itex]H = A_3 [/itex]. I know that the quotient group is the cosets of [itex]H[/itex], but then I realized that I have no idea how to interpret a Cartesian product of any type of set with elements that aren't just numbers. An ordered pair of permutations doesn't make sense (this is not a homework question). I'd be grateful for some clarity.
     
  2. jcsd
  3. Nov 13, 2013 #2

    Office_Shredder

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    If G1 and G2 are groups, then
    [tex] G_1 \times G_2 = \{ (g_1,g_2)\ :\ g_1 \in G_1,\ g_2\in G_2 \} [/tex]
    with the multiplication
    [tex] (g_1,g_2)\cdot (h_1,h_2) = (g_1 h_1, g_2 h_2) [/tex].

    So if you have G = S3, and H = A3, G/H is isomorphic to the two element group {1,-1} (where each permutation gets mapped to its parity), and a general element of A3 x (S3/A3) is [itex] (\sigma, \pm 1 ) [/itex] where sigma here is any even permutation.

    For example,
    [tex] \left( (1 2 3 ),-1 \right) \cdot \left( (1 2 3), 1 \right) = \left( (1 3 2), -1 \right) [/tex]
    is a multiplication inside of this group.
     
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