# Groups with abelian automorphism group

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?

• Some more details are given here. Instead of Heinken and Liebeck one could use the reference G.A. Miller. Jan 25 '16 at 12:02
• @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. Jan 25 '16 at 13:17

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).

• The questioner starts: "Given $K$ with $\lvert K\rvert \ge 5$ …". How does this fit with $K = 1$? Oct 25 '18 at 11:36
• @LSpice it fits with $K=1$ because there're no $|K|\ge 5$ assumption in the reference.
– YCor
Oct 25 '18 at 12:42