On the automorphism group of a binary $q$-analog of the Fano plane

TL;DR

This paper investigates the automorphism group of a potential binary q-analog of the Fano plane, combining theoretical and computational methods, and concludes it is either trivial or cyclic of small order.

Contribution

It provides a detailed analysis of the automorphism group structure of the binary q-analog of the Fano plane, narrowing down possible groups and offering general results for automorphisms of order 2.

Findings

Automorphism group is either trivial or cyclic of order 2, 3, or 4.

Remaining possible groups of order 2 and 4 are explicitly identified.

For automorphisms of order 2, a general result applies to any binary q-Steiner triple system.

Abstract

The smallest set of admissible parameters of a $q$-analog of a Steiner system is $S_2[2,3,7]$. The existence of such a Steiner system -- known as a binary $q$-analog of the Fano plane -- is still open. In this article, the automorphism group of a putative binary $q$-analog of the Fano plane is investigated by a combination of theoretical and computational methods. As a conclusion, it is either rigid or its automorphism group is cyclic of order $2$, $3$ or $4$. Up to conjugacy in $\operatorname{GL}(7,2)$, there remains a single possible group of order $2$ and $4$, respectively, and two possible groups of order $3$. For the automorphisms of order $2$, we give a more general result which is valid for any binary $q$-Steiner triple system.