Babai–Goodman bounded-orbit lattice conjecture

For every finite group $G$ , there exists a finite lattice $L$ such that $\operatorname{Aut}(L)\cong G$ and the action of $\operatorname{Aut}(L)$ on the elements of $L$ has at most $C$ orbits, where $C$ is an absolute constant independent of $G$ . Equivalently, if $\lambda(G)$ denotes the minimum number of $\operatorname{Aut}(L)$ -orbits on $L$ , taken over all finite lattices $L$ with $\operatorname{Aut}(L)\cong G$ , then $\sup_G\lambda(G)<\infty$ .$

References

Primary source

arXiv

Additional references

Bounded-orbit lattice representations of finite groups — arXiv — JiaLi Du, Andrea Lucchini, Joy Morris, Pablo Spiga

finite-group-theory lattice-theory

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A 2026 paper claims to settle the conjecture with a universal bound of 50, but the claim has not been independently corroborated.

The conjecture asks whether one universal constant bounds the number of orbits in a lattice representation of every finite group. The supplied record identifies no earlier lattice resolution.

Known results

Barmak (2020) settled the analogous poset version with four orbits, while explicitly leaving the lattice problem unresolved.

September 28, 2026 claimed resolution

JiaLi Du, Andrea Lucchini, Joy Morris, and Pablo Spiga reportedly prove $\lambda(G)\le 50$ for every finite group and arrange a regular orbit. The bound is not claimed optimal. This is a claimed complete resolution, but the supplied searches found no independent corroboration or verification.

Current status (as of September 2026): The conjecture is claimed solved with $\lambda(G)\le 50$ , but that claim remains unverified; optimality of $50$ is open.

Sources

No solutions have been posted yet.