Group

Definition

A group is an algebraic structure which is a monoid with the additional requirement of a unary operation, the inverse, which gives rise to the identity element.


Written out explicitly, a group is a set G with a binary operation ∗ satisfying the following properties:

  1. Closure: ∀a,b∈G, a∗b∈G.
  2. Associativity: ∀a,b,c∈G, a∗(b∗c)=(a∗b)∗c.
  3. Existence of an identity: ∃e∈G,∀a∈G: a∗e=e∗a=a. This element e is called the identity.
  4. Existence of an inverse: ∀a∈G,∃a′∈G:a∗a′=a′∗a=e. This element a′ is called the inverse.

Notes on Notation

In groups where the binary operation is multiplication, often a−1 is the notation used for the inverse, whereas where the underlying operation is addition, −a will often be used.

The same is true for the use of 1 and 0 for the multiplicative and additive identity respectively, while id is often used in a more general context.

Similarly na may be used to denote a∗a∗a…a with n terms if the underlying operation is addition, or an if the underlying operation is multiplication, despite the fact that in the first case multiplication need not be defined and in the second case exponentiation need not be defined.

Motivating Examples