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 (a^n = e). If no such n exists then a is said to have infinite order.
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.
