# P12863: Common divisors of disjoint coset indices

- ID: `P12863`
- Reference: `kourovka-20-99-sun-gcd-of-disjoint-coset-indices`
- Page: https://theoremdb.org/statements/P12863
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/kourovka-20-99-sun-gcd-of-disjoint-coset-indices?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let \(k>1\), let \(G\) be a group, and let \(a_1H_1,\ldots,a_kH_k\) be pairwise disjoint left cosets of subgroups of finite index in \(G\). Must there be indices \(1\leq i<j\leq k\) such that \(\gcd([G:H_i],[G:H_j])\geq k\)?

## Status

**Resolution status:** Open.

Research work is recorded. A complete resolution has not passed review.

## Research packet

### 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-20-99-sun-gcd-of-disjoint-coset-indices`, 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.