21.113 (2026)

Let $G$ be a finite group and $p$ be a prime. Let $\Psi_{p,G}$ be the class function of $G$ which vanishes on all $p$-singular elements of $G$ and whose value at each $p$-regular element $x$ of $G$ is the number of $p$-elements of $C_G(x)$.
$\qquad$ a) Is it true that $\Psi_{p,G}$ is a character of $G$?
$\qquad$ b) If yes, can $\Psi_{p,G}$ be afforded by a projective $RG$-module, where $R$ is a complete discrete valuation ring of characteristic zero such that the field of fractions of $R$ is a splitting field for $G$ and its subgroups, and the residue field $R/J(R)$ is a splitting field of characteristic $p$ for $G$ and its subgroups?

It is known that $\Psi_{p,G}$ is a character when $G \cong S_n$ for any positive integer $n$ and any prime $p$ (T. Scharf, J. Algebra, 139, no. 2 (1991), 446–457).