Abstract algebra MOC

Group theory MOC

Group theory is predominantly concerned with symmetry, and its algebraic representation as a group, the fundamental object of study.

Objects

• Group

Type of group

• Torsion group
• -group
• Nilpotent group
• Solvable group
• Abelian group
  • Elementary abelian group
• Finite group
  • Cyclic group
• Simple group
• Topological group
  • Discrete group
  • Compact group
  • Lie group

Special groups

• Symmetric group
• Alternating group
• General linear group

Morphisms

• Group homomorphism
  • Kernel of a group homomorphism
• Group action
  • Group representation theory MOC

Internally

• Conjugation by an element
  • Group class function
• Coset
• Torsion group
• Subgroup (subobjects)
  • Cyclic subgroup
  • Normal subgroup
  • Core of a subgroup

Commutation

• Centre of a group
• Centralizer in a group
• Group commutator

Externally

• Quotient group (quotient objects)
• Direct product of groups (product) Free product of groups (coproduct)
• Semidirect product

Functors in

• Free group
  • Group presentation

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