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Thread: Presentation of a group in standard form

  1. #1
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    Presentation of a group in standard form

    Hi

    I have the following group presentation
    G = a,b,c | ab = ba, ac = ca, bc = cb, a^2 = b^3 = c^3 = e.
    I need to show that all the elements of G can be written in the standard form
    a^ib^jc^k for i ∈{0,1} and j,k ∈{0,1,2}.
    I'm not really sure how to go about this so any help would be great.
    Thanks, Alex
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  2. #2
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    Re: Presentation of a group in standard form

    The fact that the given "presentation" starts "a, b, c" says that any member of G can be written as products of a, b, and c. The fact that "ab= ba, ac= ca, bc= cb" means that the order does not matter so any member is of the form "a^ib^jc^k" for some i, j, k. Use the conditions "a^2= b^3= c^3= e" to show that i need not be larger than 2 (fpr example a^5= (a^2)(a^2)(a)= (e)(e)(a)= a) and j and k need not be larger than 3.
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  3. #3
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    Re: Presentation of a group in standard form

    Thank you! That does seem to make sense.
    Would it be enough to just give one example to show that i need not be larger than 2 and j,k need not be larger than 3, or would I need a general proof to show this?
    Last edited by alexlbrown59; Nov 8th 2016 at 04:37 AM.
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  4. #4
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    Re: Presentation of a group in standard form

    No. You need to use the fact that, for any i, either i= 2n or i= 2n+ 1 and that, for any j and k, j= 3m or 3m+1 or 3m+ 2 and k= 3p or 3p+ 1 or 3p+ 2.
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  5. #5
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    Re: Presentation of a group in standard form

    Ok so I have a^2n=(a^n)^2=e, and a^2n+1=a(a^n)^2=ae=a, hence i need not be larger than 2, and as a^2=e, we have i=0 or 1.
    Is this correct? If so then I think I feel confident enough to do the same for b and c.
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  6. #6
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    Re: Presentation of a group in standard form

    Yes, that is true.
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