Definition:Group

Definition
A group is a semigroup with an identity (i.e. a monoid) in which every element has an inverse.

Group Axioms
The properties that define a group are sufficiently important that they are often separated from their use in defining semigroups, monoids, etc. and given recognition in their own right.

A group is an algebraic structure $$\left({G, \cdot}\right)$$ which satisfies the following four conditions:

These four stipulations are called the group axioms (or group postulates).

Some sources use the notation $$\left \langle G, \cdot \right \rangle$$ for $$\left({G, \cdot}\right)$$.

Group Product
The notation $$\left({G, \cdot}\right)$$ is used to represent a group whose underlying set is $$G$$ and whose operation is $$\cdot$$.

The operation $$\cdot$$ is referred to as the group product or just product.

Product Notation
When discussing a general group with a general group product, it is customary to dispense with a symbol for this operation and merely concatenate the elements to indicate the product.

That is, we invoke the product notation and write $$a b \in G$$ instead of $$a \cdot b \in \left({G, \cdot}\right)$$, as this can make the notation more compact and the arguments easier to follow.

Historical Note
The first one to formulate a set of axioms to define the structure of a group was Leopold Kronecker in 1870.

Also see

 * Abelian group