Fixed-point-free automorphism of order three implies nilpotent

Statement

Suppose $G$ is a finite group and $\varphi$ is an automorphism of $G$ of order three. Then, $G$ is a nilpotent group and $g$ commutes with $\varphi (g)$ for every $g\in G$.

Related facts

• Fixed-point-free involution on finite group is inverse map
• Fixed-point-free automorphism of prime order implies nilpotent
• Fixed-point-free automorphism of order four implies solvable

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 336, Theorem 1.5, Section 10.1 (Elementary properties)