Restriction Criteria for Endotrivial Modules

Definitions and mathematical statement

Let $p$ be a prime, let $G$ be a finite group, and let $S$ be a Sylow $p$-subgroup of $G$. The supplied statement does not specify the field $k$, its characteristic, or whether $S\ne 1$. The following is a conditional formulation of the intended modular representation setting: assume that $k$ is a field of characteristic $p$ and that $S\ne 1$. These additional hypotheses require confirmation; any intended splitting-field or algebraic-closure hypothesis is also unresolved.

For a subgroup $Q\le G$, write \[ N_G(Q)=\{g\in G:gQg^{-1}=Q\}, \] and write $H'=[H,H]$ for the commutator subgroup of a group $H$. Interpreting the source's subgroup indices as nontrivial subgroups of $S$, define, for every $1\ne Q\le S$, \[ \rho_1(Q)=N_G(Q)',\qquad \rho_i(Q)=\left\langle\bigcup_{1\ne R\le S}\left(N_G(Q)\cap\rho_{i-1}(R)\right)\right\rangle \quad(i\ge2), \qquad \rho^\infty(S)=\bigcup_{i\ge1}\rho_i(S). \] Here $\langle X\rangle$ denotes the subgroup generated by $X$. The displayed initial term and recursion are retained exactly as supplied; the source does not expressly give the domain of $Q$ or a convention for $S=1$.

All modules below are finite-dimensional left modules. A $kH$-module $M$ is endotrivial if \[ \operatorname{End}_k(M)\cong k\oplus P \] as $kH$-modules for some projective $kH$-module $P$, where $k$ is the trivial module and $H$ acts on endomorphisms by $(h\cdot f)(m)=h f(h^{-1}m)$. Write $U\simeq_{\mathrm{st}}V$ if $U\oplus P\cong V\oplus P'$ for some projective modules $P,P'$ over the same group algebra.

Supplied conjecture, subject to the qualifications above. For every one-dimensional $kN_G(S)$-module $L$, with associated character $\lambda_L:N_G(S)\to k^\times$, \[ \left(\exists\text{ an endotrivial }kG\text{-module }M: \operatorname{Res}^{G}_{N_G(S)}M\simeq_{\mathrm{st}}L\right) \quad\Longleftrightarrow\quad \rho^\infty(S)\le\ker(\lambda_L). \]

The source also calls this equivalent to saying that the image of $K(G)\to K(N_G(S))$ is the dual of $N_G(S)/\rho^\infty(S)$, but it defines neither $K$ nor the dual. Under the conventional interpretation, let $T(H)$ be the group of stable isomorphism classes of endotrivial $kH$-modules, with tensor product as its operation, and set \[ K(H)=\ker\!\left(\operatorname{Res}^{H}_{S}:T(H)\longrightarrow T(S)\right) \quad\text{for }H\in\{G,N_G(S)\}. \] The intended dual would then be \[ \operatorname{Hom}\!\left(N_G(S)/\rho^\infty(S),k^\times\right), \] viewed through the stable classes of the corresponding one-dimensional modules. With these interpretations the source's second formulation reads \[ \operatorname{im}\!\left(\operatorname{Res}^{G}_{N_G(S)}:K(G)\longrightarrow K(N_G(S))\right) = \left\{[L]\in K(N_G(S)): \dim_k L=1,\ \rho^\infty(S)\le\ker(\lambda_L)\right\}. \] The conventions for $K$, the dual, and the asserted equivalence require confirmation from the cited source.

Short English statement

The supplied conjecture asks whether a one-dimensional representation of the normalizer of a Sylow $p$-subgroup extends stably to an endotrivial representation of the whole finite group exactly when it kills the recursively defined subgroup $\rho^\infty(S)$. The field hypotheses, the trivial Sylow case, and the notation in the claimed equivalent formulation need confirmation. The supplied curation labels the problem partially solved; that assessment is not independently verified here.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json: supplied complete record with permanent ID $30002881$, title Restriction Criteria for Endotrivial Modules, and exact source code catalogue entry. The record gives proposed year $2015$ and identifies no individual proposer.
• Supplied report citation: Cohomology of Finite Groups: Interactions and Applications (2016), Oberwolfach Reports, pp. 1317--1384, DOI 10.4171/owr/2015/24. https://doi.org/10.4171/owr/2015/24. The supplied literature list also identifies this as Oberwolfach Report 2015/24.
• Supplied literature reference: Carlson--Thévenaz, The torsion subgroup of endotrivial modules. https://arxiv.org/abs/1408.3982.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports that the conjecture is proved for groups with abelian Sylow $p$-subgroups, that further special Sylow cases are known, and that no general proof is established. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; the record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication. This is a supplied editorial label, not mathematical verification of this draft or independent inspection of either reference.