Finite groups with only small automorphism orbits
Alexander Bors

TL;DR
This paper classifies finite groups with automorphism orbits of bounded size, showing specific small groups have orbits of size at most 3, infinitely many have orbits up to size 8, and larger groups are constrained by bounds related to their generators and solvability.
Contribution
The paper provides a complete classification for groups with orbits of size at most 3, constructs examples with orbits up to size 8, and establishes bounds on group order based on orbit size and number of generators.
Findings
Groups with orbit size ≤ 3 are explicitly classified.
Infinitely many groups have orbit size exactly 8.
Groups with orbit size ≤ 23 are solvable.
Abstract
We study finite groups such that the maximum length of an orbit of the natural action of the automorphism group on is bounded from above by a constant. Our main results are the following: Firstly, a finite group only admits -orbits of length at most if and only if is cyclic of one of the orders , , , or , or is the Klein four group or the symmetric group of degree . Secondly, there are infinitely many finite (-)groups such that the maximum length of an -orbit on is . Thirdly, the order of a -generated finite group such that only admits -orbits of length at most is explicitly bounded from above in terms of and . Fourthly, a finite group such that all -orbits on are of length at most …
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