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Math Help - Generators for Z/nZ

  1. #1
    Super Member Bernhard's Avatar
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    Generators for Z/nZ

    Dummit and Foote Section 1.2 Dihedral Groups Exercise 15 reads as follows:

    Find a set of generators and relations for \mathbb{Z}/n \mathbb{Z}
    ---------------------------------------------------------------------------------------------

    If you particularize the problem to say \mathbb{Z}/4 \mathbb{Z} then 1 + 4 \mathbb{Z} is a generator and so is 3 + \mathbb{Z}.

    But relations??? Are there any?

    I guess then 1 + n \mathbb{Z} is a generator for \mathbb{Z}/n \mathbb{Z}. Is that correct? Other generators? Relations???

    Can someone please clarify and help?

    Peter
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  2. #2
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    Re: Generators for Z/nZ

    Z/nZ is cyclic, so it has at least one generator, call it x. we only need to subject this generator to one relation to recover all the algebraic behavior of Z/nZ:

    x^n = e.

    (this is a really simple example, so don't over-think it).
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  3. #3
    Super Member Bernhard's Avatar
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    Re: Generators for Z/nZ

    Thanks

    You write "this is a really simple example, so don't over-think it"

    Good advice - I was looking for too much in it

    Peter
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