No 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 where G is a finite group and is an automorphism of G with a cycle of length greater than 1 2 G . In particular, it is shown that every finite group G having such an automorphism is abelian.

Call a reduced word w multiplicity-bounding if and only if a finite group on which the word map of w has a fiber of positive proportion ρ can only contain each nonabelian finite simple group S as a composition factor with multiplicity bounded in terms of ρ and S. In this paper, based on recent work of Nikolov, we present methods to show that a given reduced word is multiplicity-bounding and apply them to give some nontrivial examples of multiplicity-bounding words, such as words of the form x e , where x is a single variable and e an odd integer.

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

We study the nonabelian composition factors of a finite group G assumed to admit an Aut(G)-orbit of length at least ρ|G|, for a given ρ ∈ (0, 1]. Our main results are the following: The orders of the nonabelian composition factors of G are then bounded in terms of ρ, and if ρ > 18 19 , then G is solvable. On the other hand, for each nonabelian finite simple group S, there is a constant c(S) ∈ (0, 1] such that S occurs with arbitrarily large multiplicity as a composition factor in some finite group G having an Aut(G)-orbit of length at least c(S)|G|.