Minimal Families of Nonisomorphic Subgroup Embeddings

Definitions and mathematical statement

Retained source question. Is this family $(B;A_\lambda)$ of pairwise nonisomorphic subgroup embeddings the first family which occurs? A subgroup embedding can be represented by a pair $(B;A)$ with $A\le B$. Two such pairs are isomorphic if an isomorphism of ambient groups $\phi:B\to B'$ satisfies $\phi(A)=A'$. For fixed $B$, pairwise nonisomorphism means the $A_\lambda$ belong to distinct $\operatorname{Aut}(B)$-orbits. This defines the ordinary embedding terminology but does not reconstruct the missing family.

The ambient group or module $B$, subgroups $A_\lambda$, parameter set, prime and exponent, and the ordering measured by first are absent. The supplied assessment mentions the classical ${\mathbb{Z}}/p^n$ setting, but does not say whether minimality concerns exponent, ambient type, ambient order, subgroup type, or another complexity. Nor is family defined as a parameterized construction across primes or rings, a positive-dimensional family over an infinite field, or a finite collection for fixed data. These possibilities cannot be identified silently; for a fixed finite group the number of subgroups is finite. Thus no precise minimality proposition can be extracted from the surviving question alone.

Conflicting supplied status notes. The triage checked 2026-08-21 labels the item open and says checked literature did not settle minimality, despite finite-type lower bounds and related examples. The later statement-verification field instead reports that the official report's final theorem answers this very minimal-family question positively, labels the statement unrecoverable, and expressly declines to promote a separate clean open statement. The theorem and the family are not reproduced. This draft retains that reported positive answer and the conflicting dated triage as provenance; it does not invent an open conjecture, a theorem statement, or a recovered family.

Short English statement

The question concerns minimality of a particular family of nonisomorphic subgroup embeddings, but the family and the meaning of minimality are missing. A later verification note reports a positive answer in the source itself, conflicting with the dated open-status triage. No separate clean open problem is reconstructed.

Sources

• Local UnsolvedMath registry, permanent ID 30000184, code catalogue entry; Minimal Families of Nonisomorphic Subgroup Embeddings, Oberwolfach Reports Open Problems. Statement status unrecoverable; no clean statement supplied.
• Representation Theory of Finite-Dimensional Algebras (2005), Oberwolfach Reports, DOI 10.4171/owr/2005/06, pp. 319--406. https://doi.org/10.4171/owr/2005/06. Proposed year 2005; no proposer named.
• Supplied literature label Schmidmeier, subgroup embeddings, original OWR report: https://doi.org/10.4171/owr/2005/06. The final theorem reported to answer the question positively is not included in this record.
• Literature triage checked 2026-08-21, registry status open; record updated 2026-09-16. The conflicting later verification claim was not independently checked here.