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Math Help - Numbers, Symmetries and Groups Question

  1. #1
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    Numbers, Symmetries and Groups Question

    Many Thanks in advance, I dont know where to start on this one

    Which of the following sets are groups with respect to matrix multiplication
    a) real 2
    2 matrices with positive determinant.
    b) invertible real 2
    2 matrices of the form
    / a b \
    \ c d /

    In each of the cases below decide whether the set
    S is a subgroup of the group G.
    c) G = GL2(R) and S the set of all matrices in G of the form / a b \
    \ -b a /
    d)
    G the group S8 and S the set of all permutations in S8 that fix at least 3 numbers.
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  2. #2
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    Quote Originally Posted by callumh167 View Post
    Many Thanks in advance, I dont know where to start on this one

    Which of the following sets are groups with respect to matrix multiplication
    a) real 2 2 matrices with positive determinant.
    b) invertible real 2 2 matrices of the form
    / a b \
    \ c d /


    In each of the cases below decide whether the set
    S is a subgroup of the group G.


    c) G = GL2(R) and S the set of all matrices in G of the form / a b \
    \ -b a /


    d)
    G the group S8 and S the set of all permutations in S8 that fix at least 3 numbers.


    In order to be a group you must verify associativity of multiplication, identify the identity matrix, and then determine that each element of the group has an inverse under matrix multiplication. You probably could just assume associativity of multiplication, since you are given that from G. You will also take the identity from G. The difficult part is finding an inverse for each element of S and verifying that the inverse is in S.

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  3. #3
    MHF Contributor
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    For subgroups of G, the identity matrix is \left[\begin{array} {cc} 1 & 0 \\ 0 & 1 \end{array} \right]

    Since only matrices with determinant zero are not invertible, you know that all matrices with positive determinant are invertible. Also, you will make use of {det(A)}\cdot{det(B)} = det(AB).
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