Classification of Finite Group Automorphisms with a Large Cycle

Abstract

The investigation of finite groups classifying as “extreme” according to certain quantitative conditions has a long history. The aim of this paper is to classify the pairs (G, α) where G is a finite group and α is an automorphism of G with a cycle of length greater than . In particular, it is shown that every finite group G having such an automorphism is abelian.

Key Words:

• Automorphisms
• Cycle structure
• Finite groups
• Large cycles

2010 Mathematics Subject Classification:

• Primary: 20B25
• 20G40
• 20D45
• 20K01
• 20K30
• Secondary: 11A07
• 12E20
• 15A21
• 37P99

ACKNOWLEDGMENTS

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