20Dxx Abstract finite groups
This subtopic studies abstract finite groups, including classification themes, solvability, and the structural properties of finite group systems.
Specific topics
20D05 Finite simple groups and their classification
Overview
20D05 studies finite simple groups and their classification in abstract finite groups. It studies the structure of finite groups through subgroup structure, solvability, nilpotence, and classification techniques.
Related Wikipedia Page
Abstract Finite Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for finite simple groups and their classification
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze finite symmetry groups and their internal composition.
Applications
- Finite-group classification
- Subgroup structure
- Automorphisms and arithmetic properties
References
Recommended Textbooks
20D06 Simple groups: alternating groups and groups of Lie type
Overview
20D06 studies simple groups: alternating groups and groups of lie type in abstract finite groups. It studies the structure of finite groups through subgroup structure, solvability, nilpotence, and classification techniques.
Related Wikipedia Page
Abstract Finite Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for simple groups: alternating groups and groups of lie type
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze finite symmetry groups and their internal composition.
Applications
- Finite-group classification
- Subgroup structure
- Automorphisms and arithmetic properties
References
Recommended Textbooks
20D08 Simple groups: sporadic groups
Overview
20D08 studies simple groups: sporadic groups in abstract finite groups. It studies the structure of finite groups through subgroup structure, solvability, nilpotence, and classification techniques.
Related Wikipedia Page
Abstract Finite Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for simple groups: sporadic groups
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze finite symmetry groups and their internal composition.
Applications
- Finite-group classification
- Subgroup structure
- Automorphisms and arithmetic properties
References
Recommended Textbooks
20D10 Solvable groups, theory of formations, Schunck classes
Overview
20D10 studies solvable groups, theory of formations, schunck classes in abstract finite groups. It studies the structure of finite groups through subgroup structure, solvability, nilpotence, and classification techniques.
Related Wikipedia Page
Abstract Finite Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for solvable groups, theory of formations, schunck classes
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze finite symmetry groups and their internal composition.
Applications
- Finite-group classification
- Subgroup structure
- Automorphisms and arithmetic properties
References
Recommended Textbooks
20D15 Nilpotent groups, $p$-groups
Overview
20D15 studies nilpotent groups, $p$-groups in abstract finite groups. It studies the structure of finite groups through subgroup structure, solvability, nilpotence, and classification techniques.
Related Wikipedia Page
Abstract Finite Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for nilpotent groups, $p$-groups
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze finite symmetry groups and their internal composition.
Applications
- Finite-group classification
- Subgroup structure
- Automorphisms and arithmetic properties
References
Recommended Textbooks
20D20 Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure
Overview
20D20 studies sylow subgroups, sylow properties, $\pi$-groups, $\pi$-structure in abstract finite groups. It studies the structure of finite groups through subgroup structure, solvability, nilpotence, and classification techniques.
Related Wikipedia Page
Abstract Finite Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for sylow subgroups, sylow properties, $\pi$-groups, $\pi$-structure
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze finite symmetry groups and their internal composition.
Applications
- Finite-group classification
- Subgroup structure
- Automorphisms and arithmetic properties
References
Recommended Textbooks
20D25 Special subgroups (abelian, cyclic, etc.)
Overview
20D25 studies special subgroups (abelian, cyclic, etc.) in abstract finite groups. It studies the structure of finite groups through subgroup structure, solvability, nilpotence, and classification techniques.
Related Wikipedia Page
Abstract Finite Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for special subgroups (abelian, cyclic, etc.)
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze finite symmetry groups and their internal composition.
Applications
- Finite-group classification
- Subgroup structure
- Automorphisms and arithmetic properties
References
Recommended Textbooks
20D30 Series and lattices of subgroups
Overview
20D30 studies series and lattices of subgroups in abstract finite groups. It studies the structure of finite groups through subgroup structure, solvability, nilpotence, and classification techniques.
Related Wikipedia Page
Abstract Finite Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for series and lattices of subgroups
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze finite symmetry groups and their internal composition.
Applications
- Finite-group classification
- Subgroup structure
- Automorphisms and arithmetic properties
References
Recommended Textbooks
20D35 Subnormal subgroups of abstract finite groups
Overview
20D35 studies subnormal subgroups of abstract finite groups in abstract finite groups. It studies the structure of finite groups through subgroup structure, solvability, nilpotence, and classification techniques.
Related Wikipedia Page
Abstract Finite Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for subnormal subgroups of abstract finite groups
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze finite symmetry groups and their internal composition.
Applications
- Finite-group classification
- Subgroup structure
- Automorphisms and arithmetic properties
References
Recommended Textbooks
20D40 Products of subgroups of abstract finite groups
Overview
20D40 studies products of subgroups of abstract finite groups in abstract finite groups. It studies the structure of finite groups through subgroup structure, solvability, nilpotence, and classification techniques.
Related Wikipedia Page
Abstract Finite Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for products of subgroups of abstract finite groups
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze finite symmetry groups and their internal composition.
Applications
- Finite-group classification
- Subgroup structure
- Automorphisms and arithmetic properties
References
Recommended Textbooks
20D45 Automorphisms of abstract finite groups
Overview
20D45 studies automorphisms of abstract finite groups in abstract finite groups. It studies the structure of finite groups through subgroup structure, solvability, nilpotence, and classification techniques.
Related Wikipedia Page
Abstract Finite Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for automorphisms of abstract finite groups
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze finite symmetry groups and their internal composition.
Applications
- Finite-group classification
- Subgroup structure
- Automorphisms and arithmetic properties
References
Recommended Textbooks
20D60 Arithmetic and combinatorial problems involving abstract finite groups
Overview
20D60 studies arithmetic and combinatorial problems involving abstract finite groups in abstract finite groups. It studies the structure of finite groups through subgroup structure, solvability, nilpotence, and classification techniques.
Related Wikipedia Page
Abstract Finite Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for arithmetic and combinatorial problems involving abstract finite groups
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze finite symmetry groups and their internal composition.
Applications
- Finite-group classification
- Subgroup structure
- Automorphisms and arithmetic properties
References
Recommended Textbooks