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Let us say that $n = 5$. The number of generators of a finite cyclic group would be the number of numbers that are relatively prime to $n$ and the identity element.

Here, when $n=5$, the number of generators would be $2,3,5$. We would also have to include the identity element generator, $e$ for each finite cyclic group. Thus the number of generators of a cyclic group with order $5$ would be 4.

Similarly, when $n=12$, the number of relatively prime generators would be $2,3,5,7,11$ and $1$ for a total of 6.

  • I think I understand how to find the number of generators of a cyclic finite group, but I can't really explain why.

  • I also can't draw the connection to this statement:

    The order of an element in a finite cyclic group is the smallest positive integer $n$ such that $a^n=e$, denoted ord $a$.

  • How would I prove that the order of an element $a \in G$ equals the order of $G$? What am I failing to consider?


marked as duplicate by Dietrich Burde abstract-algebra Apr 9 at 16:27

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Do you mean, if $G=\langle a \rangle$, then to prove $|a|=|G|$ ? $\endgroup$ – Chinnapparaj R Apr 9 at 15:51
  • 2
    $\begingroup$ You are confusing a number of different things and you might need to improve your notation to make yourself clear. The identity element in a group is only a generator of the whole group if the group itself has order $1$ - ie it is the trivial group. You need to tell us what the numbers $2,3,5$ mean - you say these are "the number of generators" - that doesn't make much sense as you've written it - are these elements of the group, or something else? $\endgroup$ – Mark Bennet Apr 9 at 15:52
  • $\begingroup$ @MarkBennet The question says "Find the number of generators of a cyclic group having the given order 8". I would answer with "The number of generators of a cyclic group with order 8 is 4". I am sorry I can't explain much further, that is why I originally posted, to try to get better organized thoughts and lay out my thinking. $\endgroup$ – Evan Kim Apr 9 at 19:50

A cyclic group of order $n$ has $\phi(n)$ different generators, see these duplicates:

Cyclic Group Generators of Order $n$

How many generator has a cyclic group of order n?

How to find a generator of a cyclic group?


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