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General Linear Group not Abelian

  1. Feb 29, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that the general linear group GL(3,R) with matrix multiplication is not an abelian
    group.


    2. Relevant equations



    3. The attempt at a solution

    A group to be abelian we have to show that it satisfies the Commutativity.

    a*b=b*a

    How are we going to show that the general linear group GL(3,R) does not satisfies the Commutativity?
     
  2. jcsd
  3. Feb 29, 2012 #2

    Dick

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    Science Advisor
    Homework Helper

    Find two matrices that don't commute. Just try a few. Just don't make them both diagonal.
     
  4. Feb 29, 2012 #3
    Take

    A= 0 -1 0
    1 0 0
    0 0 1

    B= 1 0 0
    0 0 -1
    0 1 0

    A*B= 0 0 1
    1 0 0
    0 1 0

    B*A= 0 -1 0
    0 0 -1
    1 0 0

    So the two matrices do not commute as A*B=B*A and hence it is not abelian
     
  5. Feb 29, 2012 #4
    Take

    A=
    0 -1 0
    1 0 0
    0 0 1

    B=
    1 0 0
    0 0 -1
    0 1 0

    A*B=
    0 0 1
    1 0 0
    0 1 0

    B*A=
    0 -1 0
    0 0 -1
    1 0 0

    So the two matrices do not commute as A*B=B*A and hence it is not abelian
     
  6. Feb 29, 2012 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    That's just fine. Wasn't that easy?
     
  7. Feb 29, 2012 #6
    Yes easier than i expected...Thanks !!!
     
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