Maximality of Normalizers of Quasisimple Subgroups

Definitions and mathematical statement

Let $G$ be a classical group acting linearly on a vector space $V$ over its ground field $k$. The supplied record does not specify $k$, the dimension of $V$, the precise classical group, or the additional hypotheses intended by the phrase "Aschbacher--Scott setting." These remain unresolved parameters of the problem.

A group $S$ is quasisimple if $[S,S]=S$ and $S/Z(S)$ is a nonabelian simple group, where $Z(S)$ denotes its center. The action of $S$ on $V$ is irreducible over $k$ if $V\ne 0$ and its only $S$-invariant $k$-linear subspaces are $0$ and $V$. This does not impose absolute irreducibility.

For a proper quasisimple subgroup $S<G$ acting irreducibly on $V$, define its normalizer by \[ M=N_G(S)=\{g\in G:gSg^{-1}=S\}. \] A subgroup $M$ is maximal in $G$ if $M<G$ and every subgroup $H$ satisfying $M\le H\le G$ satisfies $H=M$ or $H=G$.

Problem as supplied. Determine necessary and sufficient conditions, within the intended Aschbacher--Scott setting, for $N_G(S)$ to be a maximal subgroup of $G$. Properness of $S$ alone does not supply the required properness condition on $N_G(S)$; that condition is part of the requested maximality.

The record describes this question as seeking a converse, but supplies neither the theorem to be reversed nor its hypotheses. Consequently, this is a characterization question with an unresolved scope, rather than a uniquely specified conjectural implication.

Short English statement

The question asks when the normalizer of an irreducibly acting quasisimple subgroup is maximal in its ambient classical group. The intended Aschbacher--Scott hypotheses, ground field, representation, and precise family of groups must be clarified before a fully specified characterization can be formulated.

Sources

• Local UnsolvedMath registry, permanent ID $30004421$, source problem code catalogue entry, title "Maximality of Normalizers of Quasisimple Subgroups," in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the proposed year is recorded as 2020.
• Supplied source citation: Mini-Workshop: Kronecker, Plethysm, and Sylow Branching Coefficients and their Applications to Complexity Theory (2021), Oberwolfach Reports, pp. 517--568, DOI 10.4171/owr/2020/9. Supplied source URL and literature-source link labelled "original OWR report": https://doi.org/10.4171/owr/2020/9. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• The supplied literature assessment was checked on 2026-08-22. This is supplied dated curation, not independently verified here. It labels the problem open and states that the extract asks generally for a converse concerning maximal subgroups without specifying a precise theorem, so no uniquely identifiable resolved claim can be assessed. The supplied curation expressly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review claiming self-containment and preservation of the mathematical target. These are supplied editorial assertions; they do not resolve the missing mathematical hypotheses or constitute independent mathematical verification.