When is Aut(G) abelian?

Question

Let $G$ be a group such that $\operatorname{Aut}(G)$ is abelian. Is then $G$ abelian?

This is a sort of generalization of the well-known exercise, that $G$ is abelian when $\operatorname{Aut}(G)$ is cyclic, but I have no idea how to answer it in general. At least, the finitely generated abelian groups $G$ such that $\operatorname{Aut}(G)$ is abelian can be classified.

Answer 1

From MathReviews:

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MR0367059 (51 #3301) Jonah, D.; Konvisser, M. Some non-abelian $p$-groups with abelian automorphism groups. Arch. Math. (Basel) 26 (1975), 131–133.

This paper exhibits, for each prime $p$, $p+1$ nonisomorphic groups of order $p^8$ with elementary abelian automorphism group of order $p^{16}$. All of these groups have elementary abelian and isomorphic commutator subgroups and commutator quotient groups, and they are nilpotent of class two. All their automorphisms are central. With the methods of the reviewer and Liebeck one could also construct other such groups, but the orders would be much larger.

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FYI, I found this via a google search.

The first to construct such a group (of order $64 = 2^6$) was G.A. Miller* in 1913. If you know something about this early American group theorist (he studied groups of order 2, then groups of order 3, then…and he was good at it, and wrote hundreds of papers!), this is not so surprising. I found a nice treatment of "Miller groups" in Section 8 of Mahalanobis - The Diffie–Hellman Key Exchange Protocol and non-abelian nilpotent groups.

(*): The wikipedia page seems a little harsh. As the present example shows, he was a very clever guy.

Answer 2

The answer is No. (Pete beat me to it.)

The earliest example seems to be in a 1913 paper of GA Miller's A non-abelian group whose group of isomorphisms is abelian, Messenger of Math. 48 (1913) 124--125.

Answer 3

Two additional remarks:

1. Any group whose automorphism group is abelian must have nilpotency class at most two, because the inner automorphism group, being a subgroup of the automorphism group, is abelian.
2. For finite groups, being abelian and the automorphism group being abelian as well implies cyclic. In the infinite case, there are locally cyclic groups that are not cyclic, and these have abelian automorphism groups. For instance, the additive group of rational numbers has an abelian automorphism group (the multiplicative group of rational numbers).

Two other references (in addition to Miller and Jonah-Konvisser mentioned above) for examples of 2-groups with abelian automorphism groups:

1. Some nonabelian 2-groups with abelian automorphism groups by Rebecca Roth Struik, Archiv der Mathematik, ISSN 1420-8938 (Online), ISSN 0003-889X (Print), Volume 39,Number 4, Page 299 - 302(Year 1982); MathReviews Number: 0684397

2. Some new non-abelian 2-groups with abelian automorphism groups by Ali-Reza Jamali, Journal of Group Theory, ISSN 14435883 (print), ISSN 14434446 (online), Volume 5,Number 1, Page 53 - 57(Year 2002); MathReviews Number: 1879516

I have more notes on groups whose automorphism group is abelian here and here.

Answer 4

A detailed survey on this problem for finite groups is now published and is available at Dattatraya Kitture and Yadav - Finite Groups with Abelian Automorphism Groups: A Survey. A pre-published version is available at https://arxiv.org/abs/1708.00615.

There have been some activities on this topic recently. No example of a non-special finite $p$-group having abelian automorphism group was know until quite recently. A class of such groups is constructed in "V. K. Jain, M. K. Yadav, On finite p-groups whose automorphisms are all central, Israel J. Math. 189 (2012), 225 – 236." This paper also contains a quick survey of results on the topic and a big bibliography. Some more, different kind of examples are available at Jain, Rai, and Yadav - On finite p-groups with abelian automorphism group.

Answer 5

Using GAP one finds that there are $589$ nonabelian groups of order $2187$ whose automorphism group is abelian. All the automorphism groups are isomorphic to $C_3^{12}$. The list of those groups is $\operatorname{SmallGroup}(2187,n)$ for

6172~6183, 6187~6189, 6191, 6195, 6221~6229, 6233~6235, 6269~6304, 6314~6352, 
6356~6363, 6365~6366, 6368~6376, 6378~6383, 6385, 6387~6391, 6393~6396, 6399, 
6567~6575, 6589, 6593~6604, 6606, 6612~6614, 6616~6624, 6630~6632, 6634~6642, 
6644~6647, 6649~6651, 6653~6655, 6657, 6660, 6663, 6666~6667, 6674~6676, 6679, 
6683~6686, 6688~6709, 6712, 6714, 6717~6718, 6722, 6725, 6728~6732, 6734, 6736, 
6738, 6740~6751, 6753~6762, 6764, 6767~6771, 6773~6775, 6777, 6779~6782, 6788~6790, 
6792~6793, 6795~6797, 6801~6802, 6804, 6808~6818, 6823~6831, 6836~6844, 6846~6853, 
6855~6864, 6866~6872, 6874~6887, 6889~6910, 6913~6914, 6916, 6919~6922, 6924, 
6926~6935, 6937~6940, 6942~6977, 6979~6981, 6983, 6985~6988, 6990, 6992~6998, 
7001, 7003~7024, 7026~7028, 7030~7032, 7034~7047, 7049~7058, 7060, 7062~7068, 
7070, 7072~7079, 7081, 7083~7085, 7087~7092, 7094~7097, 7101~7111, 7122, 7164, 
7166, 7168~7177, 7179, 7181, 7183~7192, 7194, 7196, 7198~7207.

Also, there are no nonabelian groups of order $729$ or $15625$ with abelian automorphism group.

Edit: The GAP code is

list_aut_abelian := function(n)
local m, i, G;
m := NumberSmallGroups(n);
for i in [1..m] do
G := SmallGroup(n,i);
if not IsAbelian(G) then
G := AutomorphismGroup(G);
if IsAbelian(G) then Print("Aut(SmallGroup(", n, ",", i, ") = ", 
StructureDescription(G), "\n"); fi;
fi; od; end;