# Groups represented by nonprincipal ultraproduct images

- Reference: `kourovka-21-11-groups-represented-by-nonprincipal-ultraproduct-images`
- Page: https://theoremdb.org/statements/kourovka-21-11-groups-represented-by-nonprincipal-ultraproduct-images
- Record maturity: Reviewed problem

## Problem

Can some or all groups of the following sorts be written as homomorphic images of nonprincipal ultraproducts of countable families of groups? This is Question 18 in (G. M. Bergman, Pacific J. Math., 274 (2015) 451-495). (a) Infinite finitely generated groups of finite exponent. (b) For an infinite set X, the group of those permutations of X that move only finitely many elements. It is known that no group of permutations containing an element with exactly one infinite orbit can be written as an image of such an ultraproduct (ibid.). This is Kourovka Notebook Problem \(21.11\).

## 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-21-11-groups-represented-by-nonprincipal-ultraproduct-images`, 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.