# Generating finite groups with Sylow subgroups

- Reference: `kourovka-21-26-generating-finite-groups-with-sylow-subgroups`
- Page: https://theoremdb.org/statements/kourovka-21-26-generating-finite-groups-with-sylow-subgroups
- Record maturity: Reviewed problem

## Problem

Let G be a non-trivial finite group and let p1,..., pk be the distinct prime divisors of |G|. For each i, let Hi be a Sylow pi -subgroup of G. Is it true that there exists an element x ∈ G such that for all i the subgroup Hi ∩ Hix is inclusion-minimal in {Hi ∩ Hig: g ∈ G}? This is Kourovka Notebook Problem \(21.26\).

## 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-21-26-generating-finite-groups-with-sylow-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.