Frobenius conjecture

There is another conjecture named after Frobenius called the Frobenius conjecture on nth roots. That conjecture is still open.

Statement

The Frobenius conjecture has the following equivalent formulations:

• A Frobenius kernel (proper nontrivial centrally closed normal subgroup) in a finite group must be nilpotent
• If a finite group possesses a fixed-point-free automorphism whose order is prime, then the finite group must be nilpotent

The Frobenius conjecture has been proved by Thompson in his paper Finite groups with fixed-point-free automorphisms of prime order.

References

• Normal p-complements for finite groups by John G. Thompson, Math. Zeitschr. 72, 332--354 (1960)
• Finite groups with fixed-point-free automorphisms of prime order by John G. Thompson, Proc. Nat. Acad. Sci. 45, 578-581 (1959)