# Disjoint soluble subgroups in symmetric and alternating groups

- Reference: `kourovka-21-3-disjoint-soluble-subgroups-in-symmetric-and-alternating-groups`
- Page: https://theoremdb.org/statements/kourovka-21-3-disjoint-soluble-subgroups-in-symmetric-and-alternating-groups
- Record maturity: Reviewed problem

## Problem

Let G = An or Sn and let H, K be soluble subgroups of G. For all sufficiently large n, can we always find an element x ∈ G such that H ∩ K x = 1? Does this hold for all n ⩾ 21? Note that the conclusion is false when G = S20 and H = K = (S4 ≀ S4 ) × S4. This is Kourovka Notebook Problem \(21.3\).

## 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-21-3-disjoint-soluble-subgroups-in-symmetric-and-alternating-groups`, 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.