Finite Group in Algebraic Structure

Prerequisite – Group

Finite Group:
A group of finite number of elements is called a finite group. Order of a finite group is finite.

Examples:
Consider the set, {0} under addition ({0}, +), this a finite group. In fact, this is the only finite group of real numbers under addition.

Set {1} under multiplication ({1}, *) and set {1, -1} under multiplication ({1, -1}, *) are the only finite groups of real numbers under multiplication. ({1, w, w²}, *) is also a finite group, where w and w² are imaginary cube roots of unity. ({1, -1, i, -i}, *) is a finite group, where i is square root of -1.

Now consider the set {0, 1, 2, 3} under addition modulo 4, this is a finite group. So, any set of form {0, 1, 2, …, (m-1)} under addition modulo m, is a finite group.

Consider the set {1, 3, 7, 9} under multiplication modulo 10, this is a finite group. So, any set of form S_m under multiplication modulo m, is a finite group, where, S_m is set of all Integers that are less than m and relatively prime to m.