Extensions of groups by weighted Steiner loops

TL;DR

This paper explores the extension of groups by weighted Steiner loops, providing explicit descriptions of loops with specific properties, and investigates the automorphism groups and isomorphism conditions of these extensions.

Contribution

It offers a concrete description of loop extensions by weighted Steiner loops, especially when the Steiner loop induces trivial automorphisms, and examines their automorphism groups and isomorphism conditions.

Findings

Explicit descriptions of loop extensions with weak associativity properties

Role of Fischer groups and geometry in loop extensions with right alternative property

Conditions for automorphisms and isomorphisms of the extensions

Abstract

Solving functional equations given in \cite{nagy} for extensions of a group $A$ by a weighted Steiner loop $S$ we obtain concrete description for all loops with interesting weak associativity properties if the Steiner loop $S$ induces only the trivial automorphism on $A$. We show that the restricted Fischer groups and their geometry play an important role for loop extension with right alternative property. Also the automorphism groups of these extensions as well as the conditions for isomorphisms between two extensions are studied.