A cyclic group is always commutative and may be finite or infinite.
Z4 is a cyclic group generated by 1, and it is also abelian or commutative.
A cyclic group is a group in which all the elements can be generated by a single element, called the generator. This means that if we take a particular element in a cyclic group and repeatedly apply the group operation to it with itself, we will eventually obtain all the elements in the group.
One important property of cyclic groups is that they are always abelian or commutative. This is because in a cyclic group, every element can be expressed in terms of the generator, and therefore, the group operation is commutative. For example, in the cyclic group Z6, we can choose 1 as the generator, and we have:
1+1=2
1+1+1=3
1+1+1+1=4
1+1+1+1+1=5
1+1+1+1+1+1=0
Since 1+1=2 and 2+1=3, we can see that 1+2=2+1, and similarly, for all other elements in the group. This means that the group operation is commutative or abelian.
Cyclic groups can be finite or infinite. A finite cyclic group, denoted by Zn, has n elements, while an infinite cyclic group can be generated by a single element and has an infinite number of elements. An example of a finite cyclic group is Z4, which has four elements: {0,1,2,3}. We can choose 1 as the generator, and we have the following group operation:
1+1=2
2+1=3
3+1=0
0+1=1
Therefore, Z4 is a cyclic group generated by 1, and it is also abelian or commutative.