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Math Help - Properties of groups

  1. #1
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    Properties of groups





    Can anyone help me with the above problems?
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  2. #2
    MHF Contributor kalagota's Avatar
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    3)
    (a^m)^n = \underbrace{a^m+a^m+\cdots a^m}_{n \mbox{ times}} = \underbrace{\underbrace{a+a+\cdots a}_{m \mbox{ times}}+\underbrace{a+a+\cdots a}_{m \mbox{ times}}+\cdots \underbrace{a+a+\cdots a}_{m \mbox{ times}}}_{n \mbox{ times}}

     = \underbrace{a+a+\cdots a}_{mn \mbox{ times}} = a^{mn}


    6) this has been proven many times here.. but anyways,

    let |ab|=n and assume |ba|=d. show n=d.
    suppose n>d.. (ba)^d = e \Rightarrow \underbrace{baba\cdots ba}_{d \mbox{ times}} = e

    b^{-1}\underbrace{baba\cdots ba}_{d \mbox{ times}}\underbrace{baba\cdots ba}_{n-d \mbox{ times}}b = b^{-1}\underbrace{baba\cdots ba}_{n-d \mbox{ times}}b

    e=\underbrace{aba\cdots bab}_{n \mbox{ times}}= \underbrace{aba\cdots bab}_{n-d \mbox{ times}}

    and continue....

    if n<d
    \underbrace{baba\cdots ba}_{d \mbox{ times}} = e

    \underbrace{ba\cdots b\underbrace{aba\cdots bab}_{n \mbox{ times}}a\cdots ba}_{d \mbox{ times}} = e

    babae\cdots ba=\underbrace{baba\cdots ba}_{d-n \mbox{ times}} = e

    ....
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  3. #3
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    Thanks for the reply.

    I think the equation in 3 needs multiply signs though.
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  4. #4
    MHF Contributor kalagota's Avatar
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    multiple signs? what the do you mean?
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  5. #5
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    Because (a^m)^n does not equal a^m+a^m+a^m+a^m n times
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  6. #6
    MHF Contributor kalagota's Avatar
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    ahh... yeah.. i mean, \underbrace{a^ma^ma^m\cdots a^m}_{n \mbox{ times}}... that should be it.. sorry for that..

    and, all you have to do is to remove that + signs..

    but anyways, if i assume that the operator is denoted by +, everything should be fine.. it is just a matter of how you denote your operator..
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  7. #7
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    Do you know of an alternative way of doing these proofs without using all the n times sort of things?

    It seems a bit ambiguous as it is.
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  8. #8
    MHF Contributor kalagota's Avatar
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    it's the best proof and you will see that proof in many references.. (and i can't think of another for now..)

    have you read my notes in the previous post? i edited it..
    Last edited by kalagota; March 5th 2009 at 06:15 PM.
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