Gruenberg-Kegel graphs with four edges of finite solvable cut groups

Abstract: We prove that there is a unique graph with four edges which is the Gruenberg-Kegel graph of a solvable cut group. This contributes to the classification of the Gruenberg-Kegel graphs of solvable cut groups initiated in [A. Bächle, A. Kiefer, S. Maheshwary, and A. del Río. "Gruenberg-Kegel graphs: Cut groups, rational groups and the prime graph question" Forum Mathematicum, vol. 35, no. 2, 2023, pp. 409-429. https://doi.org/10.1515/forum-2022-0086].

• The paper establishes that there exists a unique connected GK-graph with four edges for finite solvable cut groups.
• It employs minimal normal subgroup analysis, Clifford theory, and module techniques to eliminate alternate four-edge configurations.
• The study demonstrates that the realized graph emerges from specific Frobenius group constructions, advancing classification in group theory.

Unique GK-Graph with Four Edges for Solvable Cut Groups

Introduction and Context

This paper addresses the classification of Gruenberg-Kegel graphs (GK-graphs, or prime graphs) for a special class of finite groups—solvable cut groups—when these graphs have exactly four edges. The GK-graph for a finite group $G$ is the graph whose vertex set is the set of primes dividing the orders of elements of $G$, with edges between $p$ and $q$ if there exists an element of order $pq$ in $G$. This combinatorial invariant has proved useful in a range of group-theoretic investigations, especially regarding the interplay between group structure, element orders, and graph-theoretic constraints.

Cut groups, equivalently known as inverse semi-rational groups, are a notable generalization of rational groups. In finite group theory, a group is cut if each central unit of the integral group ring is trivial, or, equivalently, each $g \in G$ has every generator of $\langle g \rangle$ conjugate to $g$ or $g^{-1}$. The classification of rational and cut groups is a major theme in character theory, integral group rings, and the study of group element arithmetic properties.

Prior work has classified the GK-graphs of solvable rational and (partially) solvable cut groups. In particular, Bächle et al. (referenced as [BKMdR]) gave a nearly complete classification for solvable cut groups whose graphs have up to three vertices; for $G$0 vertices, they provided a shortlist of possible graphs (seven in total), but the realizability for four of them remained unresolved. Two of these unresolved graphs have four edges. The current paper focuses on closing these remaining gaps.

Main Results

The central theorem established is that there exists a unique connected graph with four edges that arises as the GK-graph of a solvable cut group. Explicitly, out of all possible graphs on four vertices with four edges that could theoretically appear as the GK-graph for such groups, only one is actually realized.

Formally, the main theorem states:

The only graph with four edges which is the GK-graph of a solvable cut group is the one whose edges are: $G$1, $G$2, $G$3, $G$4.

Graphically, this means among all graphs on the vertex set $G$5 with four edges, only the specific connection pattern above can occur as the GK-graph of a finite solvable cut group.

This result is sharp: several other graphs on these four vertices and four edges each (detailed in the paper as graphs (s), (t), (u), and (v)) are not realizable by any solvable cut group. The authors give explicit proofs that graphs (s) and (t) cannot arise as such GK-graphs, eliminating the last case distinctions outstanding in the existing literature.

Additionally, both the assumptions—solvability and the cut-property—are necessary for the result: there exist solvable groups (not cut) and cut groups (not solvable) realizing other four-edge graphs.

Technical Approach

The proof employs a range of modern finite group theory tools, including:

• Analysis of Minimal Normal Subgroups: The group structure is peeled back using minimality arguments, especially considering minimal normal $G$6-subgroups when the group is a semi-direct product $G$7, and leveraging properties of such modules.
• Character theory and Module Techniques: The paper uses Clifford theory, the structure of Fitting subgroups, properties of induced and restricted modules, the eigenvector property, and nilpotent group structures.
• Fusion of Graph-theoretic and Group-theoretic Constraints: For each candidate graph, the possible configurations of subgroups, normalizers, and centralizers are carefully examined against group action and module-theoretic restrictions, using both combinatorial cardinality arguments and technical lemmas about Sylow subgroups and their actions.
• Elimination via Contradiction: For each excluded graph, the argument shows that any group realizing such a graph would imply the existence of elements or subgroups violating the cut condition, rationality bounds, or the structure of feasible Fitting subgroups.

Crucial use is made of specialized results about cut groups, including structural properties from Manz and Wolf, technical constraints on element orders from prior work ([BKMdR] et al.), and module-theoretic properties ensuring the non-existence of certain configurations.

Explicit Construction and Realizability

The unique realizable graph is shown to be the GK-graph of the direct product of specific solvable cut groups: the Frobenius groups $G$8 and $G$9. The group $p$0 is rational, and $p$1 is cut; their product is readily checked to be solvable and cut, and their element order structure yields precisely the required set of edges in the GK-graph.

For the other graphs with four edges, explicit arguments are constructed to show that attempting to realize them with a solvable cut group leads to contradictions, often hinging on the non-existence of elements of certain product orders, or on fixed-point-free actions precluded by group and module structure limitations (e.g., via Frobenius kernel arguments and eigenvector properties).

Implications and Future Directions

This classification substantially advances the understanding of the relationship between the arithmetic of element orders in finite solvable cut groups and their combinatorial invariants. From a theoretical perspective, it clarifies the boundary between possible and forbidden prime graph structures in this context and strengthens the link between representation-theoretic properties of groups (e.g., rationality and cut phenomena) and their arithmetic spectra.

Practically, the result can serve to inform algorithms or classification projects where prime graph invariants are employed for recognition or exclusion of group properties or isomorphism types. It also points toward the rich interplay between modular representation theory and group algebra properties, possibly motivating further study in the structural constraints of other "rational-like" classes, such as completely real or semi-rational groups.

Potential directions for extension include a complete closed list for all numbers of edges/vertices, further exploration of connections with the spectrum of integral group rings, and applications to automorphism group recognition or computational group theory. The open cases for larger graphs, or for non-solvable cut groups, remain compelling subjects for future research.

Conclusion

The paper decisively classifies the realizable GK-graphs with four edges for solvable cut groups, showing that only one such graph can be attained, and eliminating all other possibilities via sharp group- and module-theoretic analysis. This result finalizes part of a long-standing classification problem in the arithmetic theory of finite groups and enriches the theory of cut groups and their combinatorial spectra.

---

Reference:

Gruenberg-Kegel graphs with four edges of finite solvable cut groups (2604.01817)