On automorphism groups of a biplane (121,16,2)

TL;DR

This paper investigates the automorphism groups of a hypothetical biplane with parameters (121,16,2), proving restrictions on possible automorphism orders and analyzing group actions to advance understanding of its symmetry properties.

Contribution

It proves that automorphism groups of such biplanes cannot have elements of order 11 or 13, refining the possible automorphism group structures.

Findings

Automorphism group order divides 2^7*3^2*5*7.

No automorphisms of order 11 or 13 exist.

Analyzes actions of automorphisms of order 5 or 7.

Abstract

The existence of a biplane with parameters $(121,16,2)$ is an open problem. Recently, it has been proved by Alavi, Daneshkhah and Praeger that the order of an automorphism group of a of possible biplane ${\mathcal D}$ of order $14$ divides $2^7\cdot3^2\cdot5\cdot7\cdot11\cdot13$. In this paper we show that such a biplane do not have an automorphism of order $11$ or $13$, and thereby establish that $|Aut({\mathcal D})|$ divides $2^7\cdot3^2\cdot5\cdot7.$ Further, we study a possible action of an automorphism of order five or seven, and some small groups of order divisible by five or seven, on a biplane with parameters $(121,16,2)$.