# Thread: What's the determinant of this lattice?

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

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

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

What is the determinant of $\displaystyle L$? I think the answer may be that it's the cardinality of the subset of $\displaystyle (\frac{\mathbb{Z}}{c \mathbb{Z}})^{\times}$ generated by $\displaystyle a$ and $\displaystyle b$ but I can't prove it. Is this a standard result?

2. Originally Posted by amanda19
Say $\displaystyle a$ and $\displaystyle b$ are both coprime to $\displaystyle c$. Let $\displaystyle L$ be the lattice

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

What is the determinant of $\displaystyle L$? I think the answer may be that it's the cardinality of the subset of $\displaystyle (\frac{\mathbb{Z}}{c \mathbb{Z}})^{\times}$ generated by $\displaystyle a$ and $\displaystyle b$ 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 $\displaystyle \displaystyle \frac{\mathbb{Z}}{c\mathbb{Z}}$, a weird way of writing $\displaystyle \mathbb{Z}/c\mathbb{Z}$?

3. Yes, $\displaystyle \frac{\mathbb{Z}}{c\mathbb{Z}}$ means $\displaystyle \mathbb{Z}/c\mathbb{Z}$, but I'll write $\displaystyle \mathbb{Z}/c\mathbb{Z}$ 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 )