Some discussions related to disjoint Ber's subplanes

Abstract

A projective plane ${\mathcal{P}}_{\mathcal{m}}$ is a Ber's subplane of a finite projective plane ${\mathcal{P}}_{\mathcal{n}}$ if every point and line of ${\mathcal{P}}_{\mathcal{n}}\setminus {\mathcal{P}}_{\mathcal{m}}$ is incident to some line and some point, respectively, of ${\mathcal{P}}_{\mathcal{n}}$. It is known that the order of the plane ${\mathcal{P}}_{\mathcal{n}}$ and its Ber's subplane ${\mathcal{P}}_{\mathcal{m}}$ satisfy the equation $n=m^2$. In this article we prove some properties of finite projective planes ${\mathcal{P}}_{\mathcal{n}}$ having disjoint Ber's subplanes covering it.