# Locally finite groups with few characteristic subgroups

- Reference: `kourovka-21-39-locally-finite-groups-with-few-characteristic-subgroups`
- Page: https://theoremdb.org/statements/kourovka-21-39-locally-finite-groups-with-few-characteristic-subgroups
- Record maturity: Reviewed problem

## Problem

It is known that there exist residually finite, locally finite, characteristically simple groups with finitely many orbits under automorphisms (A. B. Apps, J. Algebra, 81 (1983), 320-339). Are there any locally finite, characteristically simple groups with finitely many orbits under automorphisms that are not residually finite? This is Kourovka Notebook Problem \(21.39\).

## 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-21-39-locally-finite-groups-with-few-characteristic-subgroups`, 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.