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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!

Jef
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    Do you mean "odd order" in the first bullet point? – Kenny Wong Feb 28 '17 at 18:56
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    The first bullet seems wrong. $\mathrm{Aut}(\mathbb{Z}_3)\cong\mathbb{Z}_3^\times\cong\mathbb{Z}_2$ has even order. – David Hill Feb 28 '17 at 19:15
  • The first bullet should have "even order" indeed. – Dietrich Burde Feb 28 '17 at 20:04
  • yes, I meant even. I will edit, thanks! – Jef Feb 28 '17 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. – Derek Holt Feb 28 '17 at 21:15
  • I would be interested! – Jef Mar 01 '17 at 17:14
  • Using GAP, I found that $\operatorname{Aut}(\operatorname{SmallGroup}(729,n))$ has order $2187$ for $n=90,92,414$, order $6561$ for $n=41,46,50,53,173,178,181,183,184,186,211,213,215,224,232,234,361,362,363,387$, order $19683$ for $n=31,69,209,351,377,467$, and order $59049$ for $n=135$. $\operatorname{Aut}(\operatorname{SmallGroup}(3125,38))$ has order $15625$. – Jianing Song Aug 24 '23 at 21:32
  • It seems that there is exactly $1$ group of order $p^5$ with an odd number of automorphisms for $p\equiv 5 \pmod 6$, and the order of its automorphism group is $p^6$. For $p\equiv 1 \pmod 6$, the number of such groups is $3$, and their automorphism order is $3p^6$. In either case, such groups has rank $2$. GAP easily checks this for $p=5,7,11,13$. – Jianing Song Aug 25 '23 at 00:25

3 Answers3

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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.

Community
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Dietrich Burde
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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.

Andreas Caranti
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  • What would be the minimal order of such a group $G$ with $p$-group automorphism group (see the reference on "minimal orders")? – Dietrich Burde Feb 28 '17 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! – Andreas Caranti Feb 28 '17 at 19:59
  • @DietrichBurde, I see we are both racing to 50k ;-) I wouldn't mind being beaten by you on the finish line. – Andreas Caranti Feb 28 '17 at 20:00
  • Well, I think your article is worth more than 50k. It is very interesting and answers the question best (+1). – Dietrich Burde Feb 28 '17 at 20:01
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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".

Chain Markov
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    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 $3p^5$. – Jianing Song Aug 25 '23 at 08:43