# Subgroup structure of finitely presented groups

- Reference: `kourovka-21-138-subgroup-structure-of-finitely-presented-groups`
- Page: https://theoremdb.org/statements/kourovka-21-138-subgroup-structure-of-finitely-presented-groups
- Record maturity: Reviewed problem

## Problem

Let G be an infinite finitely presented group such that every subgroup of infinite index is free. Must G be isomorphic to either a free group or a surface group? This is Kourovka Notebook Problem \(21.138\).

## 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-21-138-subgroup-structure-of-finitely-presented-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.