# Existence of group $G$ with $|G|>2$ and automorphism group of odd order

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.

• Do you mean "odd order" in the first bullet point? Commented Feb 28, 2017 at 18:56
• The first bullet seems wrong. $\mathrm{Aut}(\mathbb{Z}_3)\cong\mathbb{Z}_3^\times\cong\mathbb{Z}_2$ has even order. Commented Feb 28, 2017 at 19:15
• The first bullet should have "even order" indeed. Commented Feb 28, 2017 at 20:04
• yes, I meant even. I will edit, thanks!
– Jef
Commented Feb 28, 2017 at 20:04
• 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. Commented Feb 28, 2017 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.

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.

• What would be the minimal order of such a group $G$ with $p$-group automorphism group (see the reference on "minimal orders")? Commented Feb 28, 2017 at 19:56
• 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! Commented Feb 28, 2017 at 19:59
• @DietrichBurde, I see we are both racing to 50k ;-) I wouldn't mind being beaten by you on the finish line. Commented Feb 28, 2017 at 20:00
• Well, I think your article is worth more than 50k. It is very interesting and answers the question best (+1). Commented Feb 28, 2017 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".

• In fact the smallest nontrivial complete group of odd order has order $352947=3\times 7^6$, as stated in the Remark after the statement of the main theorem in Complete groups of order $3p^6$. For every prime $p\equiv 1\pmod 3$, there exists a complete group of order $3p^6$, and it occurs as the automorphism group of a group of order $3p^5$. Commented Aug 25, 2023 at 8:43