Odd order groups with an automorphism cubing many elements

Abstract

We determine the structure of a nonabelian group G of odd order such that some automorphism of G sends exactly (l/p)|G| elements to their cubes, where p is the smallest prime dividing \G\. These groups are close to being abelian in the sense that they either have nilpotency class 2 or have an abelian subgroup of index p.

Classification of Finite Group Automorphisms with a Large Cycle

“…If k 1 ≥ 2 (and hence d 1 = 1), we find that ord = 2 log 2 k 1 , which becomes too small for k 1 ≥ 4. On the other hand, for k 1 = 2 or k 1 = 3, we do obtain automorphisms of Z/2Z 2 and Z/2Z 3 , respectively, with -value precisely 1 2 , as described in the first two points from Case 3 above. This concludes the analysis of the special case.…”

“…This gives us examples of finite groups with -value in 1 2 1 . In this paper, we will provide a complete answer to the following question: Question 1.1.3.…”

“…Which are the finite groups G with G > 1 2 ? That is, which finite groups admit an automorphism with a cycle of length greater than 1 2 G ?…”

“…This bound is sharp, since there exist finite nonabelian groups whose -value equals 1 2 , for example finite dihedral groups. Using Theorem 1.1.7 as well as methods for discussing automorphisms with a "large" cycle in finite abelian groups, we will be able to show the classification result Corollary 1.1.8 below.…”

“…which are products of one FDG of type (1), say of Frobenius type d 1 d r , with one FDG of type (2), (3), or (4) such that the -values of the two factors are coprime and the product of the two -values is greater than 1 2 ; that product value then also is the -value of the FDG product. As for -maximality: If G p denotes the Sylow p-subgroup of G for the unique odd prime divisor p of G , set…”