A group is a set, *G*, together with an operation • (called the *group law* of *G*) that combines any two elements *a* and *b* to form another element, denoted *a* • *b* or *ab*. To qualify as a group, the set and operation, (*G*, •), must satisfy four requirements known as the *group axioms*

**Closure**

For all *a*, *b* in *G*, the result of the operation, *a* • *b*, is also in *G*.

**Associativity**

For all *a*, *b* and *c* in *G*, (*a* • *b*) • *c* = *a* • (*b* • *c*).

**Identity element**

There exists an element *e* in *G* such that, for every element *a* in *G*, the equation *e* • *a* = *a* • *e* = *a* holds. Such an element is unique, and thus one speaks of *the*identity element.

**Inverse element**

For each *a* in *G*, there exists an element *b* in *G*, commonly denoted *a*^{−1} (or −*a*, if the operation is denoted "+"), such that *a* • *b* = *b* • *a* = *e*, where *e* is the identity element.

The result of an operation may depend on the order of the operands. In other words, the result of combining element *a* with element *b* need not yield the same result as combining element *b* with element *a*; the equation

*a* • *b* = *b* •

**Abelian Group**

An abelian group is a set, *A*, together with an operation • that combines any two elements *a* and *b *to form another element denoted *a* • *b*. The symbol • is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, (*A*, •), must satisfy five requirements known as the *abelian group axioms*:

**Closure**

For all *a*, *b* in *A*, the result of the operation *a* • *b* is also in *A*.

**Associativity**

For all *a*, *b* and *c* in *A*, the equation (*a* • *b*) • *c* = *a* • (*b* • *c*) holds.

**Identity element**

There exists an element *e* in *A*, such that for all elements *a* in *A*, the equation *e* • *a* = *a* • *e* = *a*holds.

**Inverse element**

For each *a* in *A*, there exists an element *b* in *A* such that *a* • *b* = *b* • *a* = *e*, where *e* is the identity element.

**Commutativity**

For all *a*, *b* in *A*, *a* • *b* = *b* • *a*.

A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group"