Groups

The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms.

Main classes of groups:

  • permutation groups
  • matrix groups
  • transformation groups
  • abstract groups (opposite: concrete groups)
    • quotient group (of a group G by a normal subgroup)
    • class groups of algebraic number fields
    • finite groups, periodic groups, simple groups, solvable groups, etc
  • Groups with additional structure (If the group operations m (multiplication) and i (inversion) are compatible with corresponding structure, that is, they are continuous, smooth or regular (in the sense of algebraic geometry) maps, then G is a topological group, a Lie group, or an algebraic group.)

Branches:

  • Finite group theory
  • Representation of groups
  • Lie theory
  • Combinatorial and geometric group theory

Applications:

  • Galois theory
  • Algebraic geometry
  • Algebraic number theory
  • Harmonic analysis
  • Combinatorics
  • Statistical mechanics
  • Cryptography

Want to read: wikipedia: list of group theory topics

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