2019/10/24 by Bors, Alexander
#20F24 #FOS: Mathematics #Group Theory (math.GR) #Primary: 20D45. Secondary: 20D60
paper · doi:10.48550/arxiv.1910.11145
We study finite groups G such that the maximum length of an orbit of the natural action of the automorphism group Aut(G) on G is bounded from above by a constant. Our main results are the following: Firstly, a finite group G only admits Aut(G)-orbits of length at most 3 if and only if G is cyclic of one of the orders 1, 2, 3, 4 or 6, or G is the Klein four group or the symmetric group of degree 3. Secondly, there are infinitely many finite (2-)groups G such that the maximum length of an Aut(G)-orbit on G is 8. Thirdly, the order of a d-generated finite group G such that G only admits Aut(G)-orbits of length at most c is explicitly bounded from above in terms of c and d. Fourthly, a finite group G such that all Aut(G)-orbits on G are of length at most 23 is solvable.