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.

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

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.

Compare with additive notation.

Also denoted as
Some sources use the notation $\left \langle{G, \circ}\right \rangle$ for $\left({G, \circ}\right)$.

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

 * Definition:Group Axioms/Left
 * Definition:Group Axioms/Right


 * Equivalence of Axiom Schemata for Groups


 * Definition:Abelian Group