2007/10/23 by Peter Hegarty, Hegarty, Peter
Computer Science · Mathematics · #11B25 #20D45 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Limits and Structures in Graph Theory #Number Theory (math.NT) #math.GR #math.NT #msc:11B25 #msc:20D45
paper · pdf · doi:10.48550/arxiv.0710.4289
20 pages, no figures
arxiv created 2007/10/23 · openalex publication_date 2007/10/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense, close to abelian. We prove two theorems. In the first, we completely classify all finite groups with an automorphism cubing more than half their elements. All such groups are either nilpotent class 2 or have an abelian subgroup of index at most 2. For our second theorem we show that, if a group possesses an automorphism sending more than 4/15 of its elements to their cubes, then it must be solvable. The group A5 shows that this result is best possible. Both our main findings closely parallel results of previous authors on finite groups possessing an automorphism which inverts many group elements. The technicalities of the new proofs are somewhat more subtle, and also throw up a nice connection to a basic problem in combinatorial number theory, namely the study of subsets of finite cyclic groups which avoid non-trivial solutions to one or more translation invariant linear equations.