Question I — s it true that any normal subgroup is commensurable with such a subgroup?

Definitions and mathematical statement

For subgroups $A,B\leq G$, say that $A$ and $B$ are commensurable if \[ [A:A\cap B]<\infty\quad\text{and}\quad[B:A\cap B]<\infty. \] Question I asks whether every normal subgroup is commensurable with "such a subgroup." The ambient group and the preceding family referred to by "such" are absent from the supplied excerpt.

To record the exact logical shape without inventing the missing objects, let $G$ stand for that source ambient group and $\mathcal S$ for that source-defined family. The intended assertion has the form \[ \forall N\triangleleft G\quad\exists H\in\mathcal S:\quad [N:N\cap H]<\infty,\quad[H:N\cap H]<\infty. \] This is a conditional schema for the excerpt, not a conjecture for arbitrary groups and arbitrary families. No finite-generation, finite-index, or nontriviality restriction on $N$ is supplied. A determinate mathematical problem requires recovering $G$, $\mathcal S$, and any scope restrictions from the preceding source text.

The source says that the problem was suggested by a discussion with H. Bass. The registry labels it related to the congruence subgroup property, but that label does not define the missing family or establish that it consists of congruence subgroups.

Short English statement

Is every normal subgroup of the omitted ambient group commensurable with a subgroup from the omitted preceding family? The excerpt supplies commensurability but not the objects needed to identify the conjecture.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $11000096$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Question I, PDF page 82: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Supplied access date: 2026-07-29; extraction: pdf-font-structure; page-disambiguated numbering among 329 headings.
• Source attribution: the problem was suggested by a discussion with H. Bass. No preceding subgroup construction or additional authoritative citation is supplied.