Open research questions in Finite Group Theory Research

What the literature leaves open

• The lack of a complete classification of block-transitive t-designs with λ ≤ 5. The need for a method to determine the possible parameters (v, b, r, k) of these designs.

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

• The gap in current research is the lack of characterization of finite groups with SS-quasinormality of a maximal chain of Sylow subgroups. The paper identifies the need for new characterizations of finite p-nilpotent groups and supersolvable groups.

  SS-Quasinormal Subgroups and p-Nilpotency of Finite Groups · 2026 · DOI

• The lack of understanding of the solvabilizer number of finite groups is identified. The need for further research on the involutionary solvabilizer number is noted.

  Solvabilizer numbers of finite 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

• 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

• The weight spaces of VΛ24 are well-studied objects, but extracting the 75-dimensional representation Hphys from them remains an open problem. The passage from the modular subgroup to the full Conway group remains to be computed. Two most critical open problems: 1.

  Modular Spectral Selection in the Leech Lattice: A Non-Hermitian Mechanism via Brauer Character Theory · 2026 · DOI

• The classification of axial algebras of Jordan type η = 1/2 is still an open problem. The paper aims to tackle this problem by classifying near-solid lines in Fischer spaces.

  Automorphism groups of Matsuo algebras · 2026 · DOI

• Further research could involve exploring the relationship between invariant transversals and commutator subgroups in more general cases. The authors suggest that the identification of counterexamples may inform the development of new conjectures or theorems.

  A note on invariant transversals for normal subgroups · 2026 · DOI

• The paper identifies a gap in the existing literature, as the conjecture has not been thoroughly tested. The authors note that the conjecture is false in certain cases, highlighting the need for further research.

  A note on invariant transversals for normal subgroups · 2026 · DOI

• The lack of construction of new families of Neumaier graphs with small coherent rank. The need for new techniques to study the properties of Neumaier graphs. The gap in current research on the relationship between Neumaier graphs and strongly regular graphs.

  Neumaier graphs from cyclotomy with small coherent rank · 2026 · DOI

• Future research could explore the applications of multi-EGS groups. The authors' work could be extended to other classes of self-similar groups. Further study could investigate the properties of contracting groups.

  Liftability and contracting property of multi-EGS groups · 2026 · DOI

• The lack of understanding of the N-prime graph and its connection to the group structure. The lack of progress on the Subgroup Isomorphism Problem for finite solvable groups.

  The N‐prime graph and the Subgroup Isomorphism Problem · 2026 · DOI

• The Farrell-Jones conjecture is a prominent open conjecture in algebraic and differential topology. There is a lack of results on the fibred Farrell-Jones conjecture for relatively hyperbolic groups and their automorphisms.

  Automorphisms of relatively hyperbolic groups and the Farrell–Jones conjecture · 2026 · DOI

• The problem of determining the structure of finite groups with a large normalized sum of element orders remains open. Prior work has not established a complete understanding of the ψ-value of finite groups.

  Finite groups with a large normalized sum of element orders · 2026 · DOI

• The lack of a general upper bound for the derived length of a Sylow subgroup in terms of the number of irreducible characters of degrees divisible by a prime p. The need to reduce the proof of the main theorem to proving Theorem 2.

  Sylow subgroups and the number of irreducible characters of degrees divisible by a prime p · 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.

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

• Further analysis of the solvabilizer number of finite groups is suggested. Investigation of the involutionary solvabilizer number for other groups is proposed.

  Solvabilizer numbers of finite groups · 2026 · DOI

• The authors aim to provide sufficient conditions for liftable groups and compute contracting nuclei.

  Liftability and contracting property of multi-EGS groups · 2026 · DOI

• To study the properties of the 15 non-trivial t-designs with λ ≤ 5. To extend the classification to other types of combinatorial designs.

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

• 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

• 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

• Investigating Neumaier graphs with higher coherent ranks - Constructing new families of Neumaier graphs using different methods - Studying the properties of Neumaier graphs in relation to other graph theory…

  Neumaier graphs from cyclotomy with small coherent rank · 2026 · DOI

Most-cited papers in Finite Group Theory Research

• Intersectional Invisibility: The Distinctive Advantages and Disadvantages of Multiple Subordinate-Group Identities · Sex Roles · 2008 · 1,167 citations
• The group polarization phenomenon. · Psychological Bulletin · 1976 · 692 citations
• Group Theory and its Applications to Physical Problems · American Journal of Physics · 1962 · 619 citations
• Functional Roles of Group Members · Journal of Social Issues · 1948 · 437 citations
• A Comparison of Individuals and Small Groups in the Rational Solution of Complex Problems · The American Journal of Psychology · 1932 · 217 citations
• A Comparison of Impulsive and Instrumental Subgroups of Batterers · Violence and Victims · 1998 · 190 citations
• The Structure of Unilineal Descent Groups1 · American Anthropologist · 1953 · 182 citations
• What is the Probability that Two Group Elements Commute? · American Mathematical Monthly · 1973 · 155 citations
• Identifying cohesive subgroups · Social Networks · 1995 · 134 citations
• Simultaneous group and individual centralities · Social Networks · 1991 · 125 citations

Most recent work

• Boxicity and cubicity of a subclass of divisor graphs and power graphs of cyclic groups · Discrete Mathematics · 2026
• Finite groups with invariant TI-subgroups or σ-subnormal subgroups · Journal of Algebra and Its Applications · 2026
• Towards the inductive McKay–Navarro condition for groups of Lie type · Mathematische Zeitschrift · 2026
• On the 𝐾-theory of groups of the form ℤⁿ⋊ℤ/𝕞 with 𝕞 square-free · Proceedings of the American Mathematical Society · 2026
• Constructions of minimally t-tough regular graphs · Discrete Mathematics · 2026
• Sylow subgroups for distinct primes and intersection of nilpotent subgroups · Journal of Algebra · 2026
• Groups with conjugacy classes of coprime sizes · Bulletin of the London Mathematical Society · 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

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