Kourovka Notebook Problem 15.29

Definitions and mathematical statement

For a prime number $p$, a finite $p$-group is a group $G$ of order $p^n$ for some integer $n\geq 0$; it is nontrivial when $n\geq 1$. Let $\operatorname{Aut}(G)$ denote the group of all group automorphisms of $G$, with composition as its operation. Write \[ D_8=\langle r,s\mid r^4=s^2=1,\ srs^{-1}=r^{-1}\rangle \] for the dihedral group of order $8$. The supplied statement gives $D_8\cong\operatorname{Aut}(D_8)$ as its motivating example.

Does there exist a prime $p$, an integer $n\geq 1$, and a finite group $G$ such that \[ |G|=p^n,\qquad G\not\cong D_8,\qquad G\cong\operatorname{Aut}(G)? \] Here $p$ ranges over all primes, since the supplied question specifies no particular prime. The statement parenthetically names A. Mann, whereas the background and proposer field name Ya. G. Berkovich; the supplied record does not explain their respective roles.

Short English statement

Is there a nontrivial finite group of prime-power order, other than the dihedral group of order $8$, that is isomorphic to its own automorphism group?

Sources

• Local UnsolvedMath registry, permanent ID $701117$, exact source problem code catalogue entry, title Kourovka Notebook Problem 15.29; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 15.29, 15th issue (2002); the supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors. This bibliographic identification is supplied source information and has not been independently verified here.
• Attribution as supplied: the statement begins with "(A. Mann)", while the background and structured proposer field name Ya. G. Berkovich and give the proposal year as 2002. Their respective roles are not clarified.
• The local registry labels this problem open; that label is supplied curation, not an independently verified assessment of the literature.