Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $G$ be a cyclic group. There's a theorem which states that if $|G|$ is a prime, then every non-identity member of $G$ is a generator.

What about a cyclic group whose order is not prime: Is there such a group whose every non-identity member is a generator?

Are there other necessary/sufficient conditions regarding groups whose every non-identity member is a generator? (Beyond primality of $|G|$.)

share|cite|improve this question
The statement should be that every non-identity element is a generator, since the identity element of a group is a generator if and only if the group is trivial. – Zev Chonoles Jun 7 '11 at 21:39
@Zev: Thanks, corrected. – M.S. Dousti Jun 8 '11 at 1:54
up vote 12 down vote accepted

To answer your first question: no, a cyclic group whose order is not prime must contain non-identity (thanks Zev!) elements that are not generators. Let $G$ be a cyclic group, let $g$ be any generator of $G$, and let $n$ be the order of $G$. Then for any $d$ that divides $n$, the subgroup generated by $g^d$ is not all of $G$ (this subgroup has $n/d$ elements, but $G$ has $n$ elements).

To answer your second question: for every non-identity element of $G$ to be a generator, $G$ must be a cyclic group with prime order. If $G$ weren't a cyclic group, then $G$ wouldn't have any generators at all (the definition of "cyclic group" is "a group that can be generated by a single element"), and the answer to your first question shows that the order of $G$ must be prime.

share|cite|improve this answer
+1: Nice answer! I deleted mine because appealing to Cauchy is clearly overkill for this... – t.b. Jun 7 '11 at 21:46
+1. Great answer, simple and to the point! – M.S. Dousti Jun 8 '11 at 1:56

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.