Cyclic Group

A cyclic group includes a generator, call it g. For any element x in the group, gx is also is in the the group.

g –> gg –> ggg –> gggg -> ….

At some point we complete a cycle and the next element is the identity element. We use ‘e’.

We can build a cyclic group with the imaginary number, i.

• i
• ii= -1
• iii = -i
• iiii = 1

C={1, i, -1, -i}

Notice that it took 4 iterations to generate the Identity Element.

Notice that there are four elements in the group.

For our example, the group order is 4. Officially, the Group Order is the number of elements in the group.

Important– a group doesn’t have to be cyclic to have a group order.