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.

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.

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

 * Abelian group