# Group Theory — Order of an Element in the group, generator and Cyclic Group

**Order of an element in a group**

Let G be a Group with respect to Operator * *(here a group is a set of elements which follows Closure property with *, it also obeys associativity with * , has an identity element and corresponding inverse element as well). Order is defined*** ∀ a∈ G **as

**O(a)**, where

**O(a) = n**, that is the smallest power of

**a**for which we get

**e**, the identity element

**. If no such n exists then a is said to have infinite order.**

*(a^n = e)***Note :**

- Here a^n is not equal to a multiplied to itself n times. Power is in general with respect to the operator specified, be it plus minus or multiply etc.
- Order of an identity element of the group is always 1.
- Order of an element and it’s inverse is the same.

**Solved Problem**

**Generator Element**

An element of group set is called generator element if it can express the group elements as a combination of (under the group operation) itself. In simplified terms a generator can generate all the other elements of group when the group operation is applied to it. A group can have a set of Generator elements. In the above example, 1 can generate 1,2,3,0 and 3 can generate 3,2,1,0. So both 1 and 3 can give the entire group set. So the generator set is {1,3}.

**Cyclic Group**

It is a branch of abstract algebra. A group is said to be cyclic if there is at least one generator element in it. In the above problem we have two generator elements {1,3}, so the above set on the given operation is a Cyclic group.