# 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.

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**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, w^{2}}, *) is also a finite group, where w and w^{2} 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.