Open research questions in Finite Group Theory Research

What the literature leaves open

• Further study of the properties of super M p-groups. Investigation of the relationship between super M p-groups and other types of groups. Exploration of applications of the results to other areas of mathematics.

  Some remarks on super $$M_{p}$$-groups · 2026 · DOI

• The problem of determining whether a normal subgroup of an M-group is always an M-group is a well-known gap in the literature. The introduction of super M p-groups addresses this gap for groups of odd order.

  Some remarks on super $$M_{p}$$-groups · 2026 · DOI

• To extend the proof of the refinement to other types of finite groups. To study the implications of the refinement for the representation theory of finite groups.

  McKay bijections and character degrees · 2026 · DOI

• The McKay conjecture has been one of the central problems in the representation theory of finite groups for nearly 60 years. A new refinement of the McKay conjecture is needed to provide a deeper understanding of the relationship between a finite group and its Sylow normalizer.

  McKay bijections and character degrees · 2026 · DOI

• The lack of understanding of power simple groups. The need for a characterization of solvable power simple groups. The need for a determination of a special case of non-solvable power simple groups.

  On finite power simple groups · 2026 · DOI

• Further study of the properties of Butson Hadamard matrices. Application of the new necessary condition to other problems in mathematics and computer science. Construction of new matrices using cyclotomic cosets and other techniques.

  A nonexistence criterion and new constructions for Butson Hadamard matrices · 2026 · DOI

• The fundamental question of characterizing the pairs (n, q) for which BH(n, q) matrices exist remains open. Prior work has focused on sporadic constructions and nonexistence criteria, but a systematic understanding is lacking.

  A nonexistence criterion and new constructions for Butson Hadamard matrices · 2026 · DOI

• Future research could aim to find a sharp bound for the metric for V. Future research could apply the technique involving clusters to other areas of research.

  A note on the metric of Thompson’s group V · 2026 · DOI

• The lack of a result for three conjugacy classes in prior work. The need for a constructive algorithm that, given g ∈ G, outputs ci ∈ Ci such that c1c2c3 = g.

  Products of three conjugacy classes in the alternating group · 2026 · DOI

• Future research can focus on applying the paper's algorithms to compute generalized root systems of Nichols algebras of diagonal type and of contragredient Lie superalgebras. Future research can also focus on studying the properties of the generalized root systems introduced in the paper. Future research can also explore the implications of the paper's results for the study of quantum groups and Hopf algebras.

  Computing finite Weyl groupoids · 2026 · DOI

• The classical notion of root systems is not sufficient for studying more general types of objects, such as Nichols algebras of diagonal type and contragredient Lie superalgebras. There is a need for a generalized notion of root systems that can be applied to these more general types of objects. The paper aims to fill this gap by introducing a new concept of root systems for groupoids.

  Computing finite Weyl groupoids · 2026 · DOI

• understanding equivariant analogs of rationality and stable rationality in geometry over nonclosed fields, - the unirationality of certain actions remains open

  Equivariant unirationality of Fano threefolds · 2026 · DOI

• The paper identifies a gap in the existing literature on Thompson's groups. The gap is the lack of a sharp bound for the metric for V.

  A note on the metric of Thompson’s group V · 2026 · DOI

• Further study of the products of conjugacy classes in the alternating group. Application of the result to simple groups of Lie type.

  Products of three conjugacy classes in the alternating group · 2026 · DOI

• the unirationality of certain actions remains open, - the paper does not consider all possible cases

  Equivariant unirationality of Fano threefolds · 2026 · DOI

• We settle this open question in the stronger form that, for every $x\in G$, $(G,\cB_k^x)$ can be a nontrivial $2$-design only if $G$ is an elementary abelian $p$-group.

  When Do Subset Sums in Finite Abelian Groups Support $2$-Designs? · 2026

• The study suggests that future research should investigate the development of interventions aimed at improving group cohesion and reducing conflict. The research highlights the need for further exploration of the concept of social defenses and its relation to group dynamics.

  On Fusion and Individuation Processes in Small Groups · 1983 · DOI

• The study identifies a gap in the understanding of the influence of social structure and personal boundaries on group involvement and disengagement. The research highlights the need for further investigation into the role of nonverbal behaviors in regulating intimacy, fusion, and differentiation in self-other relations.

  On Fusion and Individuation Processes in Small Groups · 1983 · DOI

• The challenge of finding a suitable time for all members. The risk of potential members seeking individual therapy instead of group therapy. The need for flexibility in group therapy scheduling.

  The Setting of the Group · 1983 · DOI

• The paper identifies a gap in understanding the optimal approach to scheduling group therapy sessions. The author questions the effectiveness of requiring total submission to the group's schedule.

  The Setting of the Group · 1983 · DOI

• Dyer's conjecture is a major open conjecture in the field. The structure of the extended weak order for Coxeter groups is not well understood.

  Extended weak order for the rank 3 universal Coxeter group · 2026 · DOI

• The classification of 2-transitive groups [19] is relied upon but the paper does not explore whether similar results could be obtained for other classes of groups beyond the sporadic groups considered.

  Sporadic groups as block-transitive automorphism groups of \( t\)-designs with \(\lambda\le5\) · 2026 · DOI

• The paper introduces the number j_k of conjugacy classes of subgroups of index p^(b-k) in P as a key tool for analyzing Alternating Group actions, and mentions the lower bound |Syl_p S_n| ≥ n-2, but no systematic study connects these tools to rank bounds across all rank-unbounded families or provides a unified computational framework for determining j_k across different group types.

  Characterising locally finite groups satisfying the strong Sylow Theorem for the prime p – Mathematics edition – Second edition · 2026 · DOI

• The bound for PSL Groups is stated as 'much better' than the Alternating Groups bound (both using similar functional form with p-unique subgroups), but no comparative analysis, improvement techniques, or optimization strategy is provided. The paper asks 'how we could improve the bound for the Alternating Groups by new ideas or could even optimise it' without proposing specific mathematical approaches or structural properties to exploit.

  Characterising locally finite groups satisfying the strong Sylow Theorem for the prime p – Mathematics edition – Second edition · 2026 · DOI

• The paper identifies a need for a systematic study of Sylow's theorems and their applications. The study aims to fill the gap by providing a clear understanding of the theoretical results.

  Sy low's Theorems And Their Applications · 2026 · DOI

Most-cited papers in Finite Group Theory Research

• Group Representation for Even and Odd Involutive Commutative Residuated Chains · Studia Logica · 2022 · 9 citations
• Alienation of Drygas’ and Cauchy’s Functional Equations · Annales Mathematicae Silesianae · 2021 · 2 citations
• Crypto-automorphism group of some quasigroups · Discussiones Mathematicae - General Algebra and Applications · 2023 · 2 citations
• Finite groups with invariant TI-subgroups or σ-subnormal subgroups · Journal of Algebra and Its Applications · 2026 · 1 citations
• On the $2$-class group of some number fields with large degree · Archivum Mathematicum · 2021 · 1 citations
• Cyclic congruences of slim semimodular lattices and non-finite axiomatizability of some finite structures · Archivum Mathematicum · 2022 · 1 citations
• A Generalization of G-Nilpotent Units in Commutative Group Rings to Direct Product Groups · Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi · 2022 · 1 citations
• Conditions for matchability in groups and field extensions II · Discussiones Mathematicae - General Algebra and Applications · 2024 · 1 citations
• Finite groups whose order graph is C4-free · Proceedings of the Estonian Academy of Sciences · 2025 · 1 citations
• Finite groups whose coprime graph is split, threshold, chordal, or a cograph · Proceedings of the Estonian Academy of Sciences · 2024 · 1 citations

Most recent work

• Finite groups with invariant TI-subgroups or σ-subnormal subgroups · Journal of Algebra and Its Applications · 2026
• Finite Non-solvable Groups Without Elements of Order 10 · Communications in Mathematics and Statistics · 2026
• ON THE RELATIONSHIP BETWEEN THE MINIMUM IRREDUCIBLE CHARACTER CODEGREE AND FINITE SIMPLE GROUPS · Bulletin of the Australian Mathematical Society · 2026
• Sporadic groups as block-transitive automorphism groups of \( t\)-designs with \(\lambda\le5\) · Ars Combinatoria · 2026
• Characterising locally finite groups satisfying the strong Sylow Theorem for the prime p – Mathematics edition – Second edition · Journal of Mathematical & Computer Applications · 2026
• Coprime commutators in profinite groups · Revista Matemática Complutense · 2026
• A New Quantitative Characterization Of The Monster Group M And The Baby Monster Group B · JOURNAL OF ADVANCES IN MATHEMATICS · 2026
• On tetravalent 3-geodesic transitive graphs · Discrete Mathematics · 2026
• Sy low's Theorems And Their Applications · 2026
• On the base size and minimal degree of transitive groups · Journal of Group Theory · 2026

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