# Hopficity of restricted wreath products

- Reference: `kourovka-21-22-hopficity-of-restricted-wreath-products`
- Page: https://theoremdb.org/statements/kourovka-21-22-hopficity-of-restricted-wreath-products
- Record maturity: Reviewed problem

## Problem

Is the (standard, restricted) wreath product G ≀ H of two finitely generated Hopfian groups Hopfian? The same question where G is assumed to be abelian or nilpotent is equivalent to Kaplansky’s direct finiteness conjecture; see (H. Bradford, F. Fournier-Facio, Math. Z., 308, no. 4 (2024), Paper no. 58). This is Kourovka Notebook Problem \(21.22\).

## 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-21-22-hopficity-of-restricted-wreath-products`, 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.