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Subgroups and matrix homework

  1. May 29, 2008 #1
    http://img145.imageshack.us/img145/9528/matrixex6.jpg

    I have 3 subgroup criterion.

    For the first i have to show H is non empty, well here the matrix

    2 1
    5 3

    is an element of H, so it is none empty.

    The second I have to show that H is closed under the binary operation of GL2R. How do I do this? By definition ig g, h are elements of H, then gh is an element of H. I have no idea how to show this is true for it?

    I can find another matrix that would be in the group of H, and I can show the product of the matrices is 1, but do I have to prove it? And how?

    Thirdly I have to show the inverse of each element of H belongs to H, which is easy.
     
  2. jcsd
  3. May 29, 2008 #2

    Dick

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    Your set of matrices are all integer matrices with the lower left entry divisible by 5 and determinant 1. Dealing with the determinant 1 condition is pretty easy if you remember det(gh)=det(g)*det(h). Let g1=[[a1,b1],[5c1,d1]] and g2=[[a2,b2],[5c2,d2]]. To show closure multiply g1*g2 and show the lower left corner is still divisible by 5 and all the entries are integers. For inverses compute (g1)^(-1) and show it has the same properties.
     
  4. May 29, 2008 #3
    I think the fastest way might be to exploit that theorem that says that if it's a subset (which it is) that has the property that [itex]a,b \in H \Rightarrow ab^{-1} \in H[/itex] then H is a subgroup. I say that because just exploiting the fact that the determinant is a homomorphism, it's not hard to show that ab^{-1} is in H without doing alot of math.
     
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