# Amenable torsion-free and prime-exponent Tarski monsters

- Reference: `kourovka-21-72-amenable-torsion-free-and-prime-exponent-tarski-monsters`
- Page: https://theoremdb.org/statements/kourovka-21-72-amenable-torsion-free-and-prime-exponent-tarski-monsters
- Record maturity: Reviewed problem

## Problem

We say that a group is a Tarski monster if it is finitely generated, not cyclic, and all of its proper non-trivial subgroups are isomorphic to each other. a) Do there exist amenable torsion-free Tarski monsters? b) Do there exist amenable Tarski monsters of prime exponent? This is Kourovka Notebook Problem \(21.72\).

## 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-21-72-amenable-torsion-free-and-prime-exponent-tarski-monsters`, 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.