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