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Math Help - Set of Generators and Relations

  1. #1
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    Set of Generators and Relations

    Find a set of generators and relations for Z/nZ (integers modulo n).

    I think it is <a, b | a^n = b^n = 1, ab=ba> but I am not positive. Would this be correct?
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Re: Set of Generators and Relations

    Quote Originally Posted by letitbemww View Post
    Find a set of generators and relations for Z/nZ (integers modulo n).

    I think it is <a, b | a^n = b^n = 1, ab=ba> but I am not positive. Would this be correct?
    No - this group is two-generated while \mathbb{Z}/n\mathbb{Z} is cyclic. Basically, you are looking for a one-generator group with the property that the generator to the power n is the identity. You do not actually need the commutator ab=ba there - this follows from the fact that your group is cyclic. So, plug these two stipulations into a presentation and see what you get...
    Last edited by Swlabr; September 5th 2011 at 02:01 AM.
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  3. #3
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    Re: Set of Generators and Relations

    Oh you're right I got mixed up. So it's just <a | a^n=1> ?
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    MHF Contributor Drexel28's Avatar
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    Re: Set of Generators and Relations

    Quote Originally Posted by letitbemww View Post
    Oh you're right I got mixed up. So it's just <a | a^n=1> ?
    Yes, this is the presentation of a cyclic group of order n.
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  5. #5
    MHF Contributor Swlabr's Avatar
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    Re: Set of Generators and Relations

    Quote Originally Posted by letitbemww View Post
    Oh you're right I got mixed up. So it's just <a | a^n=1> ?
    Indeed. Your original group was \mathbb{Z}/n\mathbb{Z}\times \mathbb{Z}/n\mathbb{Z} (the cross product of two cyclic groups of order n). Can you see why?
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