It is well known that transformations of ${\mathbb{C}}^n$ preserving the standard inner product are unitary transformations. In this paper, all bijective transformations of isotropic sets of $\mathbb{C}{P}^n$ preserving $H$-orthogonality in both directions, called $H$-orthogonal transformations, have been determined. This is a generalization of Uhlhorn's version of Wigner's unitary-antiunitary theorem. The group of $H$-orthogonal transformations on some other sets of $\mathbb{C}{P}^n$ were also determined.

Arithmetic properties of some families in ${\mathbb{F}}_l[x]/⟨x^{p^s{q}^t}-1⟩$ were obtained by using the cyclotomic classes of order 2 with respect to $n=p^s{q}^t$, where $p\equiv 3\mathrm{m}\mathrm{o}\mathrm{d}4$, $gcd(\varphi (p^s),\varphi (q^t))=2$, $l$ is a primitive root modulo $q^t$, and ${\mathrm{o}\mathrm{r}\mathrm{d}}_{p^s}(l)=\varphi (p^s)/2$. The form of these cyclotomic classes enabled us to further generalize the results obtained in ^[1]. The explicit expressions of primitive idempotents of minimal ideals in ${\mathbb{F}}_l[x]/⟨x^{p^s{q}^t}-1⟩$ were also obtained.