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Thompson proved that a finite group with a fixed point free automorphism of prime order is nilpotent. An automorphism $\phi$ of $G$ is called a splitting automorphism if for every $x\in G, xx^\phi x^{\phi^2}...x^{\phi^{n-1}}= 1$ where $n$ is the order of $\phi$. Clearly, a fixed point free automorphism of a finite group is a splitting automorphism. Kegel generalized Thompson’s result to splitting automorphisms, namely he proved that a finite group with a splitting automorphism of prime order is nilpotent. Rowley proved that a finite group with a fixed point free automorphism is solvable. However, there are non-solvable finite groups admitting a splitting automorphism. In this talk, we will prove the following result:
Theorem. A finite group with a splitting automorphism of odd order is solvable.