Kourovka Notebook Problem 21.113

Definitions and mathematical statement

Let $G$ be finite and $p$ prime. An element is $p$-regular if its order is prime to $p$, and $p$-singular otherwise. A $p$-element has order a power of $p$, including the identity of order $p^0$. Set $C_G(x)=\{g:gx=xg\}$ and define the conjugacy-invariant function \[ \Psi_{p,G}(x)= \begin{cases} |\{y\in C_G(x):|y|\text{ is a power of }p\}|, &x\text{ is }p\text{-regular},\\ 0,&x\text{ is }p\text{-singular}. \end{cases} \] (a) Is $\Psi_{p,G}$ always an ordinary character of $G$? This means the trace character of a finite-dimensional characteristic-zero representation, or equivalently a sum of irreducible complex characters with nonnegative integer coefficients.

(b) If so, can it be afforded by a finitely generated projective $RG$-module? Here $R$ is a complete discrete valuation ring of characteristic zero, with maximal ideal $J(R)$, fraction field $K$, and residue field $k=R/J(R)$ of characteristic $p$. Require $K$ and $k$ to be splitting fields for $G$ and all its subgroups: simple representations over either field remain simple after extending scalars to an algebraic closure.

A projective $RG$-module is a direct summand of a finite-rank free $RG$-module. To afford the function means that the $KG$-module $K\otimes_R P$ has trace $\Psi_{p,G}(x)$ at every $x$. The questions are universal in $G$ and $p$, with the indicated splitting-ring hypotheses in the stronger integral realization.

Short English statement

Is this centralizer-counting function always an ordinary character, and can it moreover be realized by a projective module over a splitting valuation ring?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2622 (KOU-21.113), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, version 47 (30 September 2026), new problems, Problem 21.113. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.