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Math Help - What's the determinant of this lattice?

  1. #1
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    What's the determinant of this lattice?

    Say a and b are both coprime to c. Let L be the lattice

    L = \{ (i,j) \in \mathbb{Z}^2: a^i b^j = 1 \text{ in } (\frac{\mathbb{Z}}{c \mathbb{Z}})^{\times} \}.

    What is the determinant of L? I think the answer may be that it's the cardinality of the subset of   (\frac{\mathbb{Z}}{c \mathbb{Z}})^{\times} generated by a and b but I can't prove it. Is this a standard result?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by amanda19 View Post
    Say a and b are both coprime to c. Let L be the lattice

    L = \{ (i,j) \in \mathbb{Z}^2: a^i b^j = 1 \text{ in } (\frac{\mathbb{Z}}{c \mathbb{Z}})^{\times} \}.

    What is the determinant of L? I think the answer may be that it's the cardinality of the subset of   (\frac{\mathbb{Z}}{c \mathbb{Z}})^{\times} generated by a and b but I can't prove it. Is this a standard result?
    What does any of this mean? It seems like this is bound to be an infite set, so how does one define the determinant here? What is \displaystyle \frac{\mathbb{Z}}{c\mathbb{Z}}, a weird way of writing \mathbb{Z}/c\mathbb{Z}?
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  3. #3
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    Yes, \frac{\mathbb{Z}}{c\mathbb{Z}} means \mathbb{Z}/c\mathbb{Z}, but I'll write \mathbb{Z}/c\mathbb{Z} if it's confusing. And yes, the lattice is going to be an infinite set of points (and then the determinant is defined in the standard way - see eg. Lattice (group) - Wikipedia, the free encyclopedia )
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