# Cardinality of verbal subgroups in profinite groups

- Reference: `kourovka-21-122-cardinality-of-verbal-subgroups-in-profinite-groups`
- Page: https://theoremdb.org/statements/kourovka-21-122-cardinality-of-verbal-subgroups-in-profinite-groups
- Record maturity: Reviewed problem

## Problem

Let w be a group word, and G a profinite group. Is it true that the cardinality of the set of w-values in G is either finite or at least continuum? An affirmative answer is known for several important words; see (E. Detomi, B. Klopsch, P. Shumyatsky, J. London Math. Soc. (2), 102 (2020), 977-993). This is Kourovka Notebook Problem \(21.122\).

## Status

The reviewed record remains open.

## Work

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `kourovka-21-122-cardinality-of-verbal-subgroups-in-profinite-groups`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.