Bounded-orbit lattice representations of finite groups

Abstract

For a finite group $G$, let $\lambda (G)$ denote the minimum number of orbits on the elements of a finite lattice $L$ with $\mathrm{Aut}(L)\cong G$. Babai and Goodman conjectured that $\lambda (G)$ is bounded by an absolute constant. We prove that $\lambda (G)\le 50$ for every finite group $G$, thereby confirming their conjecture. Moreover, the lattice can be chosen to have a regular orbit. The main algebraic ingredient is a decomposition of a generating set of an arbitrary finite $2$-group into an elementary abelian part and two sets in which no quotient of distinct elements is an involution.

arXiv comment13 pages