On the base size of a finite group on its action on the lattice of subgroups

TL;DR

This paper characterizes when the automorphism group of a finite group acts with a base size of one on its subgroup lattice, showing it occurs exactly when the group is cyclic.

Contribution

It proves that the automorphism group's action has base size one if and only if the finite group is cyclic, addressing a conjecture related to group representations.

Findings

Base size is 1 iff the group is cyclic.

Provides a characterization linking group structure to automorphism action.

Supports Babai's conjecture on automorphism groups of lattices.

Abstract

Given a finite group $R$, we investigate the base size of the action of the automorphism group of $R$ on the lattice of subgroups of $R$. Our main result shows that this base size is $1$ if and only if $R$ is cyclic. Our motivation arises from a conjecture of Babai on the problem of representing groups as automorphism groups of lattices with a bounded number of orbits.