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

Review:

page revision: 4, last edited: 29 Dec 2019 18:21