In Rotman's introduction to the theory of groups, there is an exercise (7.9) where one has to prove that the following facts for a finite group $G$.

  • If $G$ is abelian and $|G|>2$ then $\text{Aut}(G)$ has even order
  • $\text{Aut}(G)$ is not cyclic when $G$ is not abelian
  • $\text{Aut}(G)$ is never cyclic of odd order >1.

This is all fairly easy, but I was wondering if there even exists a group $G$ with $|G|>2$ which has automorphism group of odd order. Anyone has an example or reference (couldn't find it by simple internet search)?

Clearly $G$ can't be abelian, symmetric, alternating, dihedral, ... so the most obvious counterexamples don't work.

Thanks in advance!

  • 1
    $\begingroup$ Do you mean "odd order" in the first bullet point? $\endgroup$ – Kenny Wong Feb 28 '17 at 18:56
  • 2
    $\begingroup$ The first bullet seems wrong. $\mathrm{Aut}(\mathbb{Z}_3)\cong\mathbb{Z}_3^\times\cong\mathbb{Z}_2$ has even order. $\endgroup$ – David Hill Feb 28 '17 at 19:15
  • $\begingroup$ The first bullet should have "even order" indeed. $\endgroup$ – Dietrich Burde Feb 28 '17 at 20:04
  • $\begingroup$ yes, I meant even. I will edit, thanks! $\endgroup$ – Jef Feb 28 '17 at 20:04
  • $\begingroup$ If anyone is interested, there are also examples of complete groups of odd order (trivial centre and no outer automorphisms), but I would need to look up the references. I think the first one was found by Rex Dark. $\endgroup$ – Derek Holt Feb 28 '17 at 21:15

Yes, there exists indeed one group explicitly known with odd order automorphism group, of order $5^7$ and exponent $125$. For a discussion with more references see this MO-question.

References: On Minimal Orders of Groups with Odd Order Automorphism Groups.


There are many examples of finite $p$-groups, $p$ a prime, whose automorphism group is a $p$-group itself. So for $p > 2$ they provide examples of the kind you are looking for.

An accessible construction is given in my paper

A. Caranti. A simple construction for a class of $p$-groups with all of their automorphisms central. Rend. Semin. Mat. Univ. Padova 135 (2016), 251-258.

The paper can be found in the arXiv.

  • $\begingroup$ What would be the minimal order of such a group $G$ with $p$-group automorphism group (see the reference on "minimal orders")? $\endgroup$ – Dietrich Burde Feb 28 '17 at 19:56
  • $\begingroup$ They're biggish, at least $p^{10} \ge 3^{10}$. I was interested in a relatively straightforward construction, not in getting small groups. But of course your reference is very interesting! $\endgroup$ – Andreas Caranti Feb 28 '17 at 19:59
  • $\begingroup$ @DietrichBurde, I see we are both racing to 50k ;-) I wouldn't mind being beaten by you on the finish line. $\endgroup$ – Andreas Caranti Feb 28 '17 at 20:00
  • $\begingroup$ Well, I think your article is worth more than 50k. It is very interesting and answers the question best (+1). $\endgroup$ – Dietrich Burde Feb 28 '17 at 20:01

Actually, it is currently known that the least nontrivial odd order automorphism group has order $3^7$. It was proved by Peter Hegarty and Desmond Machale in Minimal odd order automorphism groups

Another example is definitely not minimal, but still interesting enough to be mentioned. It is a complete group of order $3\cdot 7^{12}\cdot 19$, that was constructed by R.S. Dark in "A complete group of odd order" (here is the post, from which I found out about its existence: Does every finite non-trivial complete group have even order?). It also satisfies your condition as any complete group is isomorphic to its automorphism group and moreover it serves as a counterexample to a conjecture weaker than yours - the so-called Rose conjecture, that stated, that "all nontrivial finite complete groups have even order".


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.