Frobenius conjecture
From Groupprops
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)