Sharp Group-Order Bounds from Small Brauer Degrees

Definitions and mathematical statement

Let $p$ be a prime and $G$ a finite group. For a positive integer $n$, let $n_p=p^{v_p(n)}$ denote the largest power of $p$ dividing $n$. Let $O_p(G)$ denote the largest normal $p$-subgroup of $G$, and let $\operatorname{IBr}_p(G)$ denote the set of irreducible $p$-Brauer characters of $G$, associated with simple representations over an algebraically closed field of characteristic $p$. For $\beta\in\operatorname{IBr}_p(G)$, its degree is $\beta(1)$. Let $A_7$ denote the alternating group on seven letters. Using the standard section interpretation, $A_7$ is involved in $G$ if there exist $H\leq G$ and $K\trianglelefteq H$ such that $H/K\cong A_7$.

Assume that \[ O_p(G)=1 \qquad\text{and}\qquad \beta(1)_p\leq p\quad\text{for every }\beta\in\operatorname{IBr}_p(G). \] The problem asks whether, for every such prime $p$ and finite group $G$, the following bounds hold: \[ \begin{array}{ll} \text{(i)} & |G|_2\leq 2^7,\quad\text{if }p=2;\\[2pt] \text{(ii)} & |G|_p\leq p^2,\quad\text{if }p\geq5,\text{ or if }p=3\text{ and }A_7\text{ is not involved in }G;\\[2pt] \text{(iii)} & |G|_3\leq 3^3,\quad\text{if }p=3\text{ and }A_7\text{ is involved in }G. \end{array} \] The original statement calls these conjectured sharp bounds. This is a supplied qualification, not an assertion here that the inequalities or their optimality have been established. The record supplies no extremal examples or separate precise sharpness assertion. Its abbreviated original statement refers to cases (i)--(iii) without defining them there; their conditions above come from the supplied complete and clean statements.

Short English statement

For a finite group with trivial normal $p$-subgroup, suppose every irreducible $p$-Brauer character has degree divisible by at most one factor of $p$. Must the $p$-part of the group order satisfy the stated bounds, with the bound for $p=3$ depending on whether $A_7$ occurs as a section? The supplied record labels the problem open and describes the bounds as conjecturally sharp; these labels require independent review.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json: supplied record with permanent ID $30002852$, title "Sharp Group-Order Bounds from Small Brauer Degrees," exact source code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2015.
• Supplied report citation: Representations of Finite Groups (2016), Oberwolfach Reports, DOI 10.4171/owr/2015/18, pp. 971--1027. Supplied source URL and literature-source link labelled "original OWR report": https://doi.org/10.4171/owr/2015/18. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation's year 2016 differs from the proposed year 2015 and the year component of the DOI; this discrepancy is unresolved here.
• Supplied literature source: "A note on p-parts of Brauer character degrees," https://www.cambridge.org/core/journals/bulletin-of-the-australian-mathematical-society/article/note-on-pparts-of-brauer-character-degrees/705A3F19994693902E2F2B95AD8C268B.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the problem open and states that the conjectured bounds $2^7$, $p^2$, and $3^3$ are not established in the extracted record, while later work improves some bounds without settling this exact general conjecture. The supplied background characterizes this assessment as dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The supplied registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication. This records the supplied editorial assessment, not independent mathematical or source verification by this draft.