Identify intermediate group invariants

Provide additional group invariants and determine their position relative to the partitions induced by the six group relations, specifically by identifying relations whose partitions satisfy P₁<P<P₂, P₂<P<P₅, or P₅<P<P₆.

Background

The paper compares six equivalence relations on finite groups: group isomorphism, simultaneous preservation of all power maps, isomorphism of the cyclic-subgroup semilattice, power-graph isomorphism, isomorphism of every individual power-map functional graph, and equality of element-order multisets.

The main results show that only three distinct partition levels occur among these relations for all finite groups. This problem asks whether further natural invariants can produce strictly intermediate levels between the established partitions.

References

Related to the latter, the following problem seeks to identify further relations that could lie strictly between those we have defined. Provide additional group invariants and determine their position relative to those presented in this paper. In other words, identify relations $\sim$ that generate a partition $P$ on the set of finite groups such that $P_1 < P < P_2$, $P_2<P<P_5$ or $P_5 < P < P_6$.

— On a class of combinatorial group invariants  (2609.20516 - Fernandes et al., 17 Sep 2026) in Section 6, Problem labeled \ref{prob:1}

Finally, motivated by Theorem~\ref{thm:main-alpha}, we have the following problem. Provide further classes of groups for which Theorem~\ref{thm:main} can be strengthened.

— On a class of combinatorial group invariants  (2609.20516 - Fernandes et al., 17 Sep 2026) in Section 6, final Problem