Using arbitrary bases for the finite field $\mathbb{ F}_{ q^n}$ over $\mathbb{ F}_{ q}$, we obtain the generalized M\"obius transformations (GMTs), which are a class of bijections between the projective geometry $PG(n-1,q)$ and the set of roots of unity $\mu_{ \frac{ q^n-1}{ q-1}}\subseteq\mathbb{ F}_{ q^n}$, where $n\geq 2$ is any integer. We also introduce a class of projective polynomials, using the properties of which we determine the inverses of the GMTs. Moreover, we study the roots of those projective polynomials, which lead to a three-way correspondence between partitions of $\mathbb{ F}_{ q^n}^\ast,\mu_{ \frac{ q^n-1}{ q-1}}$ and $PG(n-1,q)$. Through this correspondence and the GMTs, we construct permutation polynomials of index $\frac{ q^n-1}{ q-1}$ over $\mathbb{ F}_{ q^n}$.
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