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**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?