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arXiv · 2609.35191

Bounded-orbit lattice representations of finite groups

JiaLi Du·Andrea Lucchini·Joy Morris·Pablo Spiga
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Abstract

For a finite group $G$, let $λ(G)$ denote the minimum number of orbits on the elements of a finite lattice $L$ with $\operatorname{Aut}(L)\cong G$. Babai and Goodman conjectured that $λ(G)$ is bounded by an absolute constant. We prove that $λ(G)\leq 50$ 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.

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JiaLi Du, Andrea Lucchini, Joy Morris, Pablo Spiga. 2026-09-28. Bounded-orbit lattice representations of finite groups. https://arxiv.org/abs/2609.35191

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