Fixed-point-free automorphism of order four implies solvable

Statement

Suppose $G$ is a finite group and $\phi$ is a fixed-point-free automorphism of $G$ of order four. Then $G$ is a solvable group.

Related facts

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

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 342, Theorem 4.2, Section 10.4 (fixed-point-free automorphisms of order 4)