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 $|G|$.)

  • 5
    $\begingroup$ 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. $\endgroup$ Jun 7, 2011 at 21:39
  • $\begingroup$ @Zev: Thanks, corrected. $\endgroup$ Jun 8, 2011 at 1:54

1 Answer 1


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.

  • 4
    $\begingroup$ +1: Nice answer! I deleted mine because appealing to Cauchy is clearly overkill for this... $\endgroup$
    – t.b.
    Jun 7, 2011 at 21:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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