Tărnăuceanu–author coset-complement conjecture

For every finite group $G$ , every integer $n\ge 1$ , every choice of subgroups $H_1,\ldots,H_n\le G$ , and every $g_1,\ldots,g_n\in G$ , if $U=\bigcup_{i=1}^n g_iH_i$ is a proper subset of $G$ and $A=G\setminus U$ , then $|A|\ge \frac{|G|}{2^n}$ , equivalently $|G\setminus\bigcup_{i=1}^n g_iH_i|\ge \frac{|G|}{2^n}$ whenever the union is proper.

References

Primary source

arXiv

Additional references

On the complement of a union of cosets — arXiv

finite-groups group-theory

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

An unrefereed September 2026 preprint claims to settle the conjecture and strengthen related results, but independent verification is absent.

The conjecture concerns the complement of a union of cosets. The newly reported manuscript claims a complete resolution together with stronger structural consequences.

September 2026 preprint

The manuscript On the complement of a union of cosets claims to prove the conjectured bound, characterize equality, and strengthen the index bound for irredundant coset covers. It is explicitly unrefereed, so the claimed resolution remains unconfirmed.

Current status (as of September 2026): The conjecture is claimed solved by an unrefereed preprint, but the result remains unverified.

Sources

No solutions have been posted yet.