Kourovka Notebook Problem 15.11

Definitions and mathematical statement

For a group $G$, define its group of power automorphisms by \[ \operatorname{PAut}(G)=\{\alpha\in\operatorname{Aut}(G):\alpha(H)=H\text{ for every subgroup }H\leq G\}, \] with composition as the group operation. Thus invariance means setwise invariance, not pointwise fixation.

Is it true that for every prime $p$ and every finite abelian group $A$ of order $p^n$, where $n\geq0$ is an integer, there exists a group $G$ such that \[ \operatorname{PAut}(G)\cong A? \] No finiteness restriction on $G$ is stated in the supplied source. The case $n=0$ is included under the usual convention that the trivial group is a finite abelian $p$-group. The statement bears the attribution "M. Morigi", while the supplied proposer metadata names V. G. Bardakov; their respective roles are not explained.

Short English statement

Does every finite abelian $p$-group occur, up to isomorphism, as the full group of automorphisms preserving every subgroup of some group? The realizing group is not required to be finite in the supplied statement.

Sources

• Local UnsolvedMath registry, permanent ID $701099$, exact source code catalogue entry, title Kourovka Notebook Problem 15.11; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 15.11, 15th issue (2002), as recorded with the 21st issue edition (2026), arXiv:1401.0300v48. The collection metadata names E. I. Khukhro and V. D. Mazurov as editors.
• The statement is prefixed by "M. Morigi"; the background and proposer metadata name V. G. Bardakov, with proposed year 2002. The supplied record does not explain this attribution discrepancy.
• The local record labels the problem open and records an update date of 2026-10-16. These are supplied registry metadata, not an independently verified assessment of the literature or mathematical status. The cited source text has not been independently consulted for this draft.