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Mathematical subject classification for group theory

The Mathematical Subject Classification (MSC) is a classification scheme used for writings in mathematics. The scheme is used by the Americal Mathematical Society, as well as Zentralblatt Math. This article gives information on those aspects of the classification that are relevant for group theory.


General information about the classification

The MSC works as follows. Every article is assigned a primary classification which gives that class to which it belongs the most. Apart from this, each article may also have some secondary classification numbers, which are other classes to which it can be said to belong. These are also called its cross-references.

The classification number has three parts. The first part is a number of one or two digits, which gives the overall head within which it lies. The second part is a letter of the alphabet, which stands for a second-level classification. The third part is again a one or two-digit number, which stands for a third-level classification.

There are two main headings:

  • 20: For group theory and generalizations
  • 22: For topological groups, Lie groups

Group theory and generalizations

List of writing types

  • 00: General reference works
  • 01: Instructional exposition
  • 02: Research exposition (monograph, survey articles)
  • 03: Historical
  • 04: Explicit machine computations and programs
  • 06: Proceedings, conferences and collections

Thus, something like 20-02 means a research exposition related to group theory.

List of subtopics

  • A: Foundations
  • B: Permutation group theory
  • C: Group representation theory
  • D: Finite group theory
  • E: Structure and classification of infinite or finite groups
  • F: Special aspects of infinite or finite groups
  • G: Linear algebraic groups (classical groups)
  • H: Other groups of matrices
  • J: Connections with homological algebra and category theory
  • K: Abelian group theory
  • L05: Groupoids (small category where every morphism is an isomorphism)
  • M: Semigroup theory
  • N: Other generalizations of groups
  • P05: Probabilistic methods in group theory

A: Foundations

  • 20A05: Axiomatics and elementary properties
  • 20A10: Metamathematical considerations
  • 20A15: Applications of logic to group theory
  • 20A99: None of the above, but in this section

B: Permutation group theory

  • 20B05: General theory for finite groups
  • 20B07: General theory for infinite groups
  • 20B10: Characterization theorems
  • 20B15: Primitive groups
  • 20B20: Multiply transitive finite groups
  • 20B22: Multiply transitive infinite groups
  • 20B25: Finite automorphisms of algebraic, geometric or combiantorial structures
  • 20B27: Infinite automorphism groups
  • 20B30: Symmetric groups
  • 20B35: Subgroups of symmetric groups
  • 20B40: Computational methods
  • 20B99: None of the above, but in this section

C: Group representation theory

  • 20C05: Group rings of finite groups and their modules
  • 20C07: Group rings of infinite grousp and their modules
  • 20C08: Hecke algebras and their representations
  • 20C10: Integral representations of finite groups
  • 20C11: p-adic representations of finite groups
  • 20C12: Integral representations of infinite groups
  • 20C15: Ordinary representations and characters
  • 20C20: Modular representations and characters
  • 20C25: Projective representations and multipliers
  • 20C30: Representations of finite symmetric groups
  • 20C32: Representations of infinite symmetric groups
  • 20C33: Representations of finite groups of Lie type
  • 20C34: Representations of sporadic groups
  • 20C35: Applications of group representations to physics
  • 20C40: Computational methods
  • 20C99: None of the above, but in this section

D: Finite group theory

20D is concerned with abstract finite groups, including classification problems as well as the subgroup theory of finite groups. Third-level headings within 20D:

  • 20D05: Classification of simple and non-solvable groups
  • 20D06: Simple groups: alternating groups and groups of Lie type
  • 20D08: Simple groups: Sporadic groups
  • 20D10: Solvable groups, theory of formations, Schunk classes, Fitting classes, \pi-length, and rank
  • 20D15: Nilpotent groups, p-groups
  • 20D20: Sylow subgroups, Sylow properties, \pi-groups, \pi-structure
  • 20D25: Special subgroups (Frattini, Fitting, etc.)
  • 20D30: Series and lattices of subgroups
  • 20D35: Subnormal subgroups
  • 20D40: Products of subgroups
  • 20D45: Automorphisms
  • 20D60: Arithmetic and combinatorial problems
  • 20D69: None of the above, but in this section

E: Structure and classification of infinite or finite groups

  • 20E05: Free nonabelian groups
  • 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
  • 20E07: Subgroup theorems; subgroup growth
  • 20E08: Groups acting on trees
  • 20E10: Quasivarieties and varieties of groups
  • 20E15: Chains and lattices of subgroups, subnormal subgroups
  • 20E18: Limits, profinite groups
  • 20E22: Extensions, wreath products, and other compositions
  • 20E25: Local properties
  • 20E26: Residual properties and generalizations
  • 20E28: Maximal subgroups
  • 20E32: Simple groups
  • 20E34: General structure theorems
  • 20E36: General theorems concerning automorphisms of groups
  • 20E42: Groups with a BN-pair; buildings
  • 20E45: Conjugacy classes
  • 20E99: None of the above, but in this section

F:Special aspects of infinite or finite groups

  • 20F05: Generators, relations and presentations
  • 20F06: Cancellation theory, application of van Kampen diagrams
  • 20F10: Word problems, other decision problems, connections with logic and automata
  • 20F12: Commutator calculus
  • 20F14: Derived series, central series, and generalizations
  • 20F16: Solvable groups, supersolvable groups
  • 20F17: Formations of groups, Fitting classes
  • 20F18: Nilpotent groups
  • 20F19: Generalizations of nilpotent and solvable groups
  • 20F22: Other classes of groups defined by subgroup chains
  • 20F24: FC-groups and their generalizations
  • 20F28: Automorphism groups of groups
  • 20F29: Representations of groups as automorphisms of algebraic systems
  • 20F34: Fundamental groups and their automorphisms
  • 20F36: Braid groups, Artin groups
  • 20F38: Other groups related to topology or analysis
  • 20F40: Associated Lie structures
  • 20F45: Engel conditions
  • 20F50: Periodic groups; locally finite groups
  • 20F55: Reflection and Coxeter groups
  • 20F60: Ordered groups
  • 20F65: Geometric group theory
  • 20F67: Hyperbolic groups and nonpositively curved groups
  • 20F69: Asymptotic properties of groups
  • 20F99: None of the above, but in this section