Finite groups with only small automorphism orbits

Abstract

Abstract We study finite groups G such that the maximum length of an orbit of the natural action of the automorphism group Aut ⁢ ( G ) {\mathrm{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 ) {\mathrm{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 ) {\mathrm{Aut}(G)} -orbit on G is 8. Thirdly, the order of a d-generated finite group G such that G only admits Aut ⁢ ( G ) {\mathrm{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 ) {\mathrm{Aut}(G)} -orbits on G are of length at most 23 is solvable.