Thompson’s conjecture

The conjecture says that every finite simple nonabelian group $G$ is the product of one conjugacy class with itself: $G=C^{2}$. Thompson communicated it to Mazurov in 1982, and it appeared in Kourovka’s Notebook in 1984.

Known results

• Ellers and Gordeev (1998) reduced the remaining problem to finite simple groups of Lie type over fields of order at most $8$ and handled groups of type $\mathrm{PSL}_{n}(q)$.
• March 2024 work proved the conjecture asymptotically for all sufficiently large finite nonabelian simple groups.
• Further March 2024 bounds settled several large classical families but explicitly left finite families unresolved.

August 2026 claimed completion

Moretó and Navarro announced a manuscript claiming that all remaining Lie-type groups over fields of order at most $8$ are handled, completing the conjecture. The abstract relies on computational certificates, and no independent verification was found.

Current status (as of August 2026): The conjecture has substantial asymptotic and family-by-family results, while the claimed completion by Moretó and Navarro remains unverified.