In a paper, the authors Jonah-Konvisser say

Until recently (~1975), there were no published examples of non-abelian groups with abelian automorphism groups. Heinken and Liebeck have methods for constructing such $p$-groups;.....

When I saw the paper of Heinken and Liebeck, I saw that much of their work involves construction of following object: given a group $K$ with $|K|\geq 5$, construct a $p$-group $G$ of class $2$ and exponent $p^2$ such that $\mathrm{Aut}(G)/\mathrm{Aut}_\mathrm{c}(G)$ is isomorphic to $K$. [Here $\mathrm{Aut}_\mathrm{c}(G)$ is the set of those automorphisms of $G$ which are identity on $G/Z(G)$.]

So, I didn't find the place in which they construct groups with abelian automorphism group; this confused me with quoted statement from paper of Jonah-Konvisser.

Question: Can one indicate me where, Heinken and Liebeck do construction of non-abelian $p$-groups with abelian automorphism group?

  • $\begingroup$ Some more details are given here. Instead of Heinken and Liebeck one could use the reference G.A. Miller. $\endgroup$ Jan 25, 2016 at 12:02
  • $\begingroup$ @Burde: I am sorry. I have seen example of Miller and some generalization of it done by some people. I wanted to see the example(s) of Heinken and Liebeck; but I didn't find it explicitly in their paper, hence posted above question. $\endgroup$
    – p Groups
    Jan 25, 2016 at 13:17

1 Answer 1


With the hope that it may help somebody, I indicate that the groups $G$ with $K = 1$, the trivial group, provide the desired examples. One such graph is given in the paper itself. For more details and progress on the problem, one may look a recent survey at https://arxiv.org/abs/1708.00615, which is now published(https://link.springer.com/chapter/10.1007/978-981-13-2047-7_7).

  • 1
    $\begingroup$ The questioner starts: "Given $K$ with $\lvert K\rvert \ge 5$ …". How does this fit with $K = 1$? $\endgroup$
    – LSpice
    Oct 25, 2018 at 11:36
  • $\begingroup$ @LSpice it fits with $K=1$ because there're no $|K|\ge 5$ assumption in the reference. $\endgroup$
    – YCor
    Oct 25, 2018 at 12:42

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