$\mathbb{P}(q)$-Groupoids of Conway Type

TL;DR

This paper introduces a new class of groupoids derived from projective planes of odd order $q$, exploring their structure and the properties of their associated groups in the context of Conway's work.

Contribution

It defines $ ext{ extdollar} ext{ extbackslash mathbb{P}}(q)$-groupoids based on projective planes and investigates the structure of their associated groups, extending Conway's ideas.

Findings

Defined $ ext{ extdollar} ext{ extbackslash mathbb{P}}(q)$-groupoids from projective planes

Analyzed the structure of the associated groups

Extended Conway's framework to new algebraic objects

Abstract

In the spirit of Conway we define a groupoid starting from projective planes of order $q$, where $q$ is odd. The associated group of these groupoids is then investigated.