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Math Help - Proving groups isomorphic

  1. #1
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    Proving groups isomorphic

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

    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 \mathbb{Z}_3, 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?
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  2. #2
    tah
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    You may start with that, i.e. equating the products hk and kh, the only interesting equation you should get is b+ca'+b' = b'+c'a+b (dashed symbols are components of k) then h is in the center iff ca'=c'a holds for any a',c'\in\mathbb{Z}_3, so you deduce that c=a=0. You can see now why Z(H)\simeq \mathbb{Z}_3.

    For the second question, it's easy to check that the order of any element of H divides 3. So the order of any element of H/Z(H) divides 3 i.e. it's not cyclic. Conclude.
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