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Determine the subgroup lattice for Z8

  1. Sep 11, 2012 #1
    1. The problem statement, all variables and given/known data

    "Determine the subgroup lattice for Z8"

    2. Relevant equations

    <1>={1,2,3,4,5,6,7,0}
    <2>={2,4,6,0}
    <3>=<1>=<5>=<7>
    <4>={4,0}
    <6>={6,4,2,0}


    3. The attempt at a solution



    My book only mentions this topic in one sentence and shows a diagram for Z30, which looks like a cube.

    I don't quite see all the logic behind it. One "track" is <1> -- <3> -- <6> -- <0>. It's easy to see that each of these are subgroups of one another, but why wouldn't <1> be connected directly to <0> as well?

    My classmate told me he thinks it should look like a tetrahedron on edge.

    I'm coming up with an square based pyramid sitting upside down, with <0> as the "top" of the pyramid (but upside down). I guess it doesn't matter how it really looks. Here are the paths I have, all stemming from <1>

    One chain looks like <1> - <2> - <4> - <0>
    another looks liks <1> - <6> - <4> - <0>
    the third is <1> - <6> - <0>
    and the last is <1> - <2> - <0>



    Any advice on how to proceed would be greatly appreciated.

    Thanks!
     
    Last edited: Sep 11, 2012
  2. jcsd
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