Fusion Systems of Sectional Rank Three

Definitions and mathematical statement

Let $p$ be an odd prime. We use the standard finite-group convention for saturated fusion systems, so that $S$ is a finite $p$-group. The sectional rank of $S$ is the largest integer $r$ for which there exist $K\trianglelefteq H\leq S$ with $H/K\cong ({\mathbb{Z}}/p{\mathbb{Z}})^r$. Thus sectional rank three means that every elementary abelian section has dimension at most three over ${\mathbb{F}}_p$, and at least one has dimension exactly three.

A fusion system $\mathcal F$ on $S$ is a category whose objects are the subgroups of $S$, whose morphisms are injective group homomorphisms, and which contains all homomorphisms induced by conjugation by elements of $S$. Every morphism must factor as an isomorphism in $\mathcal F$ followed by a subgroup inclusion. Write $\operatorname{Aut}_{\mathcal F}(P)$ for the automorphism group of $P$ in this category, and $\operatorname{Aut}_S(P)$ for the subgroup induced by conjugation in $S$. Two subgroups are $\mathcal F$-conjugate if they are isomorphic in $\mathcal F$. A subgroup $P\leq S$ is fully normalized, respectively fully centralized, if $|N_S(P)|$, respectively $|C_S(P)|$, is maximal among its $\mathcal F$-conjugates.

The fusion system is saturated if the following two axioms hold. First, every fully normalized $P$ is fully centralized and $\operatorname{Aut}_S(P)$ is a Sylow $p$-subgroup of $\operatorname{Aut}_{\mathcal F}(P)$. Second, if $\varphi\in\operatorname{Hom}_{\mathcal F}(P,S)$ and $\varphi(P)$ is fully centralized, then $\varphi$ extends to a morphism $N_\varphi\to S$ in $\mathcal F$, where, writing $c_g$ for conjugation by $g$, \[ N_\varphi=\{g\in N_S(P):\varphi\circ(c_g|_P)\circ\varphi^{-1}\in\operatorname{Aut}_S(\varphi(P))\}. \]

A subgroup $Q\leq S$ is normal in $\mathcal F$ if $Q\trianglelefteq S$ and every $\varphi\in\operatorname{Hom}_{\mathcal F}(P,R)$ extends to a morphism $\overline\varphi\in\operatorname{Hom}_{\mathcal F}(PQ,RQ)$ with $\overline\varphi(Q)=Q$. The notation $O_p(\mathcal F)$ denotes the largest such subgroup. The notation $O^p(\mathcal F)$ denotes the standard $p$-residual fusion subsystem, namely the smallest normal saturated subsystem of $\mathcal F$ of $p$-power index. More explicitly, for a finite group $G$, let \[ O^p(G)=\bigcap\{N\trianglelefteq G:G/N\text{ is a }p\text{-group}\}. \] Define the hyperfocal subgroup by \[ \operatorname{hyp}(\mathcal F)=\left\langle x^{-1}\alpha(x):P\leq S,\ x\in P,\ \alpha\in O^p(\operatorname{Aut}_{\mathcal F}(P))\right\rangle. \] Here a saturated subsystem $\mathcal E$ on $T\leq S$ has $p$-power index if $\operatorname{hyp}(\mathcal F)\leq T$ and $O^p(\operatorname{Aut}_{\mathcal F}(P))\leq\operatorname{Aut}_{\mathcal E}(P)$ for every $P\leq T$.

Write $|S|=p^n$. The group $S$ has maximal nilpotency class if its nilpotency class is $n-1$, where the class is defined by the lower central series $\gamma_1(S)=S$, $\gamma_{i+1}(S)=[\gamma_i(S),S]$. A maximal subgroup of a finite $p$-group is a proper subgroup of index $p$.

Conjecture (supplied as Conjecture 1). For every odd prime $p$, every finite $p$-group $S$ of sectional rank exactly three, and every saturated fusion system $\mathcal F$ on $S$ satisfying \[ O_p(\mathcal F)=1,\qquad \mathcal F=O^p(\mathcal F), \] one has \[ \operatorname{cl}(S)=n-1 \quad\text{or}\quad \exists M\leq S\;\left([S:M]=p\ \text{and}\ M\text{ is abelian}\right). \] The alternatives are not asserted to be mutually exclusive. The superscript in $O^p(\mathcal F)$ follows the supplied corrected statement; the plain-text original does not preserve the distinction between superscripts and subscripts. This draft does not independently resolve that transcription issue against the report.

Short English statement

The conjecture concerns saturated fusion systems on finite $p$-groups of sectional rank exactly three, with $p$ odd. If the fusion system has trivial normal $p$-subgroup and equals its $p$-residual subsystem, then the underlying group should have maximal nilpotency class or an abelian subgroup of index $p$. The supplied dated curation labels the problem partially solved while retaining the full dichotomy as a conjecture; this status has not been independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006242$, title Fusion Systems of Sectional Rank Three, exact source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. The record identifies the passage as Conjecture 1 and gives proposed year 2025; no individual proposer is specified.
• Supplied source citation: Finite Groups, Fusion Systems and Applications (2025), Oberwolfach Reports, pp. 689--752, DOI 10.4171/owr/2025/16. https://doi.org/10.4171/owr/2025/16. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature-source label: "Grazian, conjecture and known cases." https://doi.org/10.4171/owr/2025/16. This is the same report link as the source citation; the supplied record does not identify an exact page for the conjecture.
• Supplied literature-source label: "Grazian, fusion systems on p-groups." https://etheses.bham.ac.uk/7670/1/Grazian17PhD.pdf. No more detailed bibliographic information is supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that the sectional-rank-three dichotomy remains conjectural in full generality, while maximal-class cases and several pearl and low-order families have been classified. This is supplied dated curation derived from the OWR material, not an independently verified assessment. Its time-sensitive status and cited claims require independent checking.
• The registry labels the statement corrected_verified and says that only formula delimiters, notation, grammar, and typography were repaired, with mathematical content and scope preserved. This is a supplied editorial label, not independent mathematical verification by this draft.