Groups

What is a Group?
A group is a combination of a set S and a binary operation '*' ( defined in any way you choose, but) with the following properties:
1)An identity element e exists,
such that for every member a of S, e * a and a * e are both identical to a.
2)Every element has an inverse: for every member a of S, there exists a member a-1 such that a * a-1 and a-1 * a are both identical to the identity element.
3)The operation is associative:
if a, b and c are members of S, then
(a * b) * c is identical to a * (b * c).
Note:
If a group is also commutative - that is, for any two members a and b of S, a * b is identical to b * a – then the group is said to be Abelian.
Examples
1)the set of integers under the operation of addition is a group.
In this group, the identity element is 0 and the inverse of any element a is its negation, -a.
The associativity requirement is met, because for any integers a, b and c, (a + b) + c = a + (b + c)
2)The nonzero rational numbers form a group under multiplication.
Here, the identity element is 1, since 1 × a = a × 1 = a for any rational number a.
The inverse of a is 1/a, since a × 1/a = 1.

3)The integers under the multiplication operation do not form a group. Because, in general, the multiplicative inverse of an integer is not an integer.
a)4 is an integer, but its multiplicative inverse is 1/4, which is not an integer.

The theory of groups is studied in group theory. A major result in this theory is the classification of finite simple groups, mostly published between about 1955 and 1983, which is thought to classify all of the finite simple groups into roughly 30 basic types.