Free Steiner triple systems and their automorphism groups

TL;DR

This paper investigates the automorphism groups of free Steiner loops, proving all automorphisms are tame and describing the structure for loops with three generators, revealing their non-finite generation in larger cases.

Contribution

It characterizes automorphism groups of free Steiner loops, showing all automorphisms are tame and providing explicit descriptions for the 3-generator case.

Findings

Automorphisms are all tame.

Automorphism group is not finitely generated for more than 3 generators.

Explicit generators and relations for the 3-generator case.

Abstract

The paper is devoted to the study of free objects in the variety of Steiner loops and of the combinatorial structures behind them, focusing on their automorphism groups. We prove that all automorphisms are tame and the automorphism group is not finitely generated if the loop is more than $3$-generated. For the free Steiner loop with $3$ generators we describe the generator elements of the automorphism group and some relations between them.