Classification of finite group automorphisms with a large cycle II

ABSTRACT

In a previous paper, the author classified the pairs (G,α) where G is a finite group and α an automorphism of G having a cycle of length greater than $\frac{1}{2}\left|G\right|$. In this paper, we extend this classification to the case where α has a cycle of length equal to $\frac{1}{2}\left|G\right|$ and show some applications of the classification: We discuss an algorithm for listing the pairs (G,α) where the largest cycle length of α equals ρ|G| on input any rational $\rho \in \left(\frac{1}{2},1\right]$, and we prove that the set of cycle length fractions of finite group automorphisms is not dense in [0,1].

KEYWORDS:

• Automorphisms
• Cycle structure
• Finite groups
• Large cycles

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20B25
• 20D45
• 20G40
• 20K01
• 20K30

Acknowledgements

The author would like to thank Peter Hellekalek, Harald Niederreiter, Alina Ostafe and Igor Shparlinski for their many helpful comments.