Sets-of-lengths isomorphism conjecture for finite groups

Prove that for any finite groups G1 and G2, equality of the collections of sets of lengths of product-one sequences, L(G1) = L(G2), implies that G1 and G2 are isomorphic.

Background

The monoid of product-one sequences over a finite group G admits factorizations into minimal product-one sequences, and L(G) denotes the collection of sets of possible factorization lengths. The authors describe a longstanding conjecture asserting that the combinatorial data L(G) determines the group up to isomorphism.

While this conjecture has seen progress for various classes of groups, it is presented here as a standing conjecture in the general setting.

References

Then, the standing conjecture is that, for finite groups $G_1$ and $G_2$, $\mathcal L (G_1) = \mathcal L (G_2)$ implies that $G_1$ and $G_2$ are isomorphic.

— A classification of finite groups with small Davenport constant (2409.00363 - Oh, 31 Aug 2024) in Section 1 (Introduction)