The binary $q$-analogue of the Fano plane has a trivial automorphism group

TL;DR

This paper proves that the binary q-analogue of the Fano plane, a specific combinatorial design, has only the trivial automorphism, resolving a key symmetry question about its structure.

Contribution

It demonstrates that the automorphism group of the binary q-analogue of the Fano plane is trivial, confirming the minimal symmetry of this combinatorial design.

Findings

Automorphism group of the binary q-analogue of the Fano plane is trivial

Supports the conjecture of minimal symmetry in such designs

Advances understanding of automorphism groups in q-analogues

Abstract

A $q$-analogue of a $t$-design is a set $S$ of subspaces (of dimension $k$) of a finite vector space $V$ over a field of order $q$ such that each $t$ subspace is contained in a constant $\lambda$ number of elements of $S$. The smallest nontrivial feasible parameters occur when $V$ has dimension $7$, $t=2$, $q=2$, and $k=3$; which is the $q$-analogue of a $2$-$(7,3,1)$ design, the Fano plane. The existence of the binary $q$-analogue of the Fano plane has yet to be resolved, and it was shown by Kiermaier et al. (2016) that such a configuration must have an automorphism group of order at most $2$. We show that the binary $q$-analogue of the Fano plane has a trivial automorphism group.