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Proving groups isomorphic

  1. Feb 27, 2009 #1
    1. The problem statement, all variables and given/known data
    Let H be the subgroup of [tex]GL(3, \mathbb{Z}_3)[/tex] consisting of all matrices of the form [tex]\left[ \begin{array}{ccc} 1 & 0 & 0 \\ a & 1 & 0 \\ b & c & 1 \end{array} \right][/tex], where [tex]a,b,c \in \mathbb{Z}_3[/tex]. I have to prove that Z(H) is isomorphic to [tex]\mathbb{Z}_3[/tex] and that [tex]H/Z(H)[/tex] is isomorphic to [tex]\mathbb{Z}_3 \times \mathbb{Z}_3[/tex].


    2. Relevant equations



    3. The attempt at a solution
    I'm really not sure how to begin with this. I started by taking two arbitrary matrices h and k from H and doing hk = kh to see what a matrix in Z(H) would have to look like, but I didn't really get anywhere with that. My initial instinct would be to just define a mapping from Z(H) to [tex]\mathbb{Z}_3[/tex], but I'm not sure how to do that, since I can't figure out what's in Z(H). Is there a better way to do this?
     
  2. jcsd
  3. Feb 27, 2009 #2

    rsg

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    These groups are known in literature as Heisenberg group over a finite field. It is generally written in the upper triangular matrix form. But that does not make any difference.
    So let me define a typical element of this field as [tex]A=\langle a,b,c\rangle[/tex] where a, b, c have the same meaning and order as you have written in the definition of the group. Let [tex]C=\langle x,y,z\rangle\in Z(H)[/tex]. Calculate [tex] AC[/tex] and [tex] CA [/tex], and equate them....


    Edited by Hurkyl: please don't give complete solutions to problems....
     
    Last edited by a moderator: Feb 27, 2009
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