A cyclic group is a group that is generated by a single element. That means that there exists an element $g$, say, such that every other element of the group can be written as a power of $g$. This element $g$ is the generator of the group.
Is that a correct explanation for what a cyclic group and a generator are? How can we find the generator of a cyclic group and how can we say how many generators should there be?
Best Answer
Finding generators of a cyclic group depends upon the order of the group. If the order of a group is $8$ then the total number of generators of group $G$ is equal to positive integers less than $8$ and co-prime to $8$. The numbers $1$, $3$, $5$, $7$ are less than 8 and co-prime to $8$, therefore if a is the generator of $G$, then $a^3,a^5,a^7$ are also generators of $G.$ Hence there are four generators of $G.$ Similarly you can find generators of groups of order $10$, $12$, $6$ etc.