**Group Theory
**

**
**Group theory is an important area of mathematics with
numerous applications, primarily in physics. Group theory belongs to a
relatively new branch of math known as abstract algebra, which deals with
abstract structures as well as operations on them.

**Definition of a
group
**

A group is a set G with a binary operation * (known as group multiplaction) on elements of G, satisfying the following four axioms:

- (Closure) G is closed with respect to multiplication, i.e. given two elements a and b of G, the product a*b also belongs to G.
- (Associativity) Given three elements a, b, and c of G, we have (a*b)*c = a*(b*c).
- (Identity) G has an identity element, denoted e, such that for every element g of G, we have g*e = e*g = g.
- (Inverse) For every
element g of G, there exists an element h of G, known as the inverse of G and
denoted g^-1, such that g*h = h*g = e.

It should be noted that the first axiom is not usually
included because closure is implied for binary operations.

**Examples of
groups
**

Although the definition of a group is abstract, there are many familiar examples of groups. Here are a few:

- The set of integers, with ordinary addition as group multiplication.
- The set of real numbers, with ordinary addition as group multiplication.
- The set of rational numbers, with ordinary addition as group multiplication.
- The set of nonzero real numbers, with ordinary multiplication as group multiplication.
- The set of nonzero rational numbers, with ordinary multiplication as group multiplication.
- The
set of integers modulo 12, with addition modulo 12 as group multiplication.

The last example may not seem familiar at first, but we in
fact use it whenever we tell time. For instance, we know that four hours after