Kourovka Notebook Problem 9.55

Definitions and mathematical statement

Let $p$ be a prime and let $G$ be a finite group of order $p^m$, where $m\geq 0$. Its integral group ring is ${\mathbb{Z}} G$, with augmentation map \[ \varepsilon:{\mathbb{Z}} G\longrightarrow{\mathbb{Z}},\qquad \varepsilon\!\left(\sum_{g\in G}a_g g\right)=\sum_{g\in G}a_g. \] For the formulation below, an augmented automorphism means a unital ring automorphism $\varphi:{\mathbb{Z}} G\to{\mathbb{Z}} G$ satisfying $\varepsilon\circ\varphi=\varepsilon$, and a central automorphism means one that fixes the center pointwise: \[ \varphi(z)=z\qquad\text{for every }z\in Z({\mathbb{Z}} G), \] where $Z({\mathbb{Z}} G)$ denotes the center of the ring. The supplied record does not define these terms; in particular, the intended meaning of "central" requires confirmation. The question below uses the stated conventions explicitly.

Define the $p$-adic group ring by \[ A=\widehat{{\mathbb{Z}}}_p G:=\varprojlim_{n\geq 1}({\mathbb{Z}}/p^n{\mathbb{Z}})G. \] A unital automorphism $\varphi$ of ${\mathbb{Z}} G$ fixes the integers and preserves each ideal $p^n{\mathbb{Z}} G$, so it induces a continuous automorphism $\widehat\varphi$ of $A$. For $\alpha\in\operatorname{Aut}(G)$, let $\alpha_*$ be the induced $\widehat{{\mathbb{Z}}}_p$-algebra automorphism of $A$, given by \[ \alpha_*\!\left(\sum_{g\in G}a_g g\right)=\sum_{g\in G}a_g\alpha(g). \] Write $A^\times$ for the group of units of $A$.

Existence question. Does there exist a prime $p$, a finite $p$-group $G$, and a central augmented automorphism $\varphi$ of ${\mathbb{Z}} G$, in the senses specified above, such that \[ \nexists\,(u,\alpha)\in A^\times\times\operatorname{Aut}(G) \quad\text{with}\quad \widehat\varphi(x)=\alpha_*(u x u^{-1})\quad\text{for every }x\in A? \] Here the displayed composition means conjugation by a unit followed by the map induced by a group automorphism, as in the supplied question. This is a statement-only formulation conditional on the terminology above; it does not resolve the source's undefined terminology.

Short English statement

Under the stated interpretation of "central augmented," the question asks whether some finite $p$-group has an augmentation-preserving automorphism of its integral group ring that fixes the ring center pointwise, but whose extension to the $p$-adic group ring cannot be obtained by conjugating by a unit and then applying a group automorphism. The source's intended terminology requires mathematical review.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $700588$, exact source problem code catalogue entry, title "Kourovka Notebook Problem 9.55." The registry labels the problem open; this is supplied metadata, not an independently verified mathematical or literature assessment.
• The supplied attribution is to K. W. Roggenkamp, proposed in 1984, in The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), original problem 9.55. The record identifies the 21st issue edition (2026), arXiv:1401.0300v48, as its source edition.
• The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors of the Kourovka Notebook collection covering Issues 1--21 (1965--2026). These bibliographic details are reproduced from the record; the cited source has not been independently consulted here.