On finite groups where the order of every automorphism is a cycle length

2014/12/29 by Bors, Alexander
#15A21 #20D15 #20D25 #20D45 #20D60 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1412.8418

Abstract

Using Frobenius normal forms of matrices over finite fields as well as the Burnside Basis Theorem, we give a direct proof of Horoševskiĭ's result that every automorphism α of a finite nilpotent group has a cycle whose length coincides with ord(α). Also, we give two new sufficient conditions for an automorphism α of an arbitrary finite group to satisfy this property, namely when ord(α) is a product of at most two prime powers or when α has a sufficiently large cycle. This will allow us to show that the least order of a group where this property is violated is 120. Finally, we observe that any finite group embeds both into a group with this property (as all finite symmetric groups enjoy the property) as well as into a finite group not having this property.

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• On the nilpotency class of finite groups with a Frobenius group of\n automorphisms
• Finite groups with only small automorphism orbits