This site will be devoted to one of the simplest algebraic structures: groups. Below is how a group is defined, along with a few basic facts and definitions.
Let G be a set of elements. While you may regard these elements as numbers, they are not necessarily numbers in the usual sense. Then G is a group if the following holds:
1) There is a binary operation 
which combines two objects into a third, such as a b = c.
2) Closure: a
b 
G.
3) Associativity: (a
b) c = a
(b c)
4) Existence of an identity: There is an element
such that a
=
a = a
5) Existence of an inverse: For each element
a there is an element 
such
that 
Properties 2 - 5 are known as the group axioms.
Basic theorems from any course or text on group theory show that the identity element is a unique element in each group, and that each element has a unique inverse element.
A subgroup is a subset of a group for
which the subset is a group in its own right. Another basic theorem
of group theory is that a nonempty subset S of G
is a subgroup if and only if for any elements a, b
S, then 
S.
Groups are isomorphic to one another if they possess all the same qualities but differ only in the names of the elements and, perhaps, the binary operation associated with each isomorphic group.
The order of a group is the number of elements in the group. If the group has infinitely many elements, then it has infinite order.
When we say "under addition" or "under multiplication," then we refer to the familiar addition or multiplication binary operations used on real numbers. Here are examples of groups which are familiar to everyone:
Complex numbers, real numbers, rational numbers, or integers under addition.
Even integers under addition are a subgroup of the integers.
Positive real numbers or positive rationals under multiplication.
The complex 
roots
of 1 form a finite group under multiplication. One of the simplest
are the four roots of 1: 
. In this example
we would say that i generates the group, which is isomorphic
to the cyclic group of order 4, which could be represented as
the rotations of a square in increments of 90 degrees.
This link will take you to a study under way of a particular class of groups which the author calls "multiplicative groups."