Kourovka Notebook Problem 19.33

Definitions and mathematical statement

Let $G$ be a finite group, let $p$ be a prime, and let $P\leq G$ be a Sylow $p$-subgroup. For a finite group $H$, let $\operatorname{Irr}(H)$ denote its irreducible ordinary complex characters, and set \[ \operatorname{Lin}(H)=\{\lambda\in\operatorname{Irr}(H):\lambda(1)=1\}. \] For $\chi\in\operatorname{Irr}(G)$ and $\lambda\in\operatorname{Lin}(P)$, the multiplicity of $\lambda$ in the restriction $\chi|_P$ is \[ m_\lambda=\langle\chi|_P,\lambda\rangle_P =\frac{1}{|P|}\sum_{x\in P}\chi(x)\overline{\lambda(x)}. \] Define the set of distinct linear constituents by \[ L_P(\chi)=\{\lambda\in\operatorname{Lin}(P):m_\lambda>0\}. \] Conjecture (E. Giannelli, Kourovka Notebook Problem 19.33). For every such $G,p,P$ and every $\chi\in\operatorname{Irr}(G)$, \[ p\mid\chi(1)\quad\text{and}\quad L_P(\chi)\ne\varnothing \quad\Longrightarrow\quad |L_P(\chi)|\geq p. \] Thus the conclusion counts distinct linear characters, regardless of their multiplicities.

The supplied source reports that the conjecture has been verified for symmetric groups, alternating groups, $p$-solvable groups, and sporadic simple groups. Here a finite group is $p$-solvable if it admits a normal series whose factors are each either of prime-power order $p^a$ or of order relatively prime to $p$. These reported verifications are source qualifications; their proofs and specific supporting references were not supplied, and they have not been independently verified here. The registry labels the general conjecture as open.

Short English statement

Giannelli's conjecture states that if an irreducible ordinary character of a finite group has degree divisible by a prime $p$, then its restriction to a Sylow $p$-subgroup has either no linear constituents or at least $p$ distinct linear constituents. The source reports verification for symmetric, alternating, $p$-solvable, and sporadic simple groups.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $701581$, exact source problem code catalogue entry, title Kourovka Notebook Problem 19.33. The supplied status is open; this is a registry label, not an independently verified literature assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 19.33, proposed by E. Giannelli in 2018. The supplied background identifies the 19th issue (2018) and the 21st issue edition (2026), with identifier arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The supplied statement reports verification for symmetric, alternating, $p$-solvable, and sporadic simple groups, without separate references for these verifications. No independent source consultation was performed.