Multiplicative loops of $2$-dimensional topological quasifields

TL;DR

This paper classifies the algebraic structure of multiplicative loops in 2-dimensional topological quasifields, focusing on those with specific subloop properties, and identifies quasifields related to certain 4-dimensional translation planes with large symmetry groups.

Contribution

It provides a detailed analysis of multiplicative loops in 2D topological quasifields and characterizes those associated with 4D translation planes with significant collineation groups.

Findings

Classification of multiplicative loops with normal subloops or compact subgroups

Explicit description of quasifields for 4D translation planes with large Lie collineation groups

Identification of algebraic structures underlying certain topological quasifields

Abstract

We determine the algebraic structure of the multiplicative loops for locally compact $2$-dimensional topological connected quasifields. In particular, our attention turns to multiplicative loops which have either a normal subloop of positive dimension or which contain a $1$-dimensional compact subgroup. In the last section we determine explicitly the quasifields which coordinatize locally compact translation planes of dimension $4$ admitting an at least $7$-dimensional Lie group as collineation group.