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.

Multiplicative 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 multiplicative 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.

Compare with additive notation.

Historical Note
The concept of the group as an abstract structure was introduced by Arthur Cayley in his 1854 paper On the theory of groups.

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