Group ↩

shorthands: {}
aliases: [group, group axioms, Group axioms]
created: 2021-10-18 15:20:49
modified: 2022-11-12 02:22:35

• (A0) (internal operation)
• (A1) associative:
• (A2) neutral element:
• (A3) inverse:

Simplest group

, and it is called the trivial group

Examples

• 3D rotations of a solid object
• with the mapping of exponentiation
• bijective continuous functions with composition as operation
• or where is a vector space and means the general linear group

Comments

• Every group is also a monoid
• The inverse of each element is unique
• There can only be one neutral element
• Commutativity is not required, a commutative group is called Abelian
• Because of A2, the empty set cannot be a group
• The neutral element is also called the identity element