2012/03/16 by Carlo Casolo, Casolo, Carlo, Orazio Puglisi +1
Computer Science · Mathematics · #20F45 #Coding theory and cryptography #Cooperative Communication and Network Coding #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #math.GR #msc:20F45
paper · pdf · doi:10.48550/arxiv.1203.3645
A mistake of th eprevious version has been corrected
openalex publication_date 2012/03/16 · arxiv created 2012/05/22 · arxiv updated 2012/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Following Plotkin we say that the automorphism x of the group G is a nil-automorphism if, for every g∈ G, there exists n=n(g) such that [g,n x]=1. If the integer n can be chosen independently of g, then x is said to be unipotent. Nil and unipotent automorphisms can be regarded as a natural extension of the concept of Engel element, since a nil-automorphism x is just a left-Engel element in G < x >. In this paper we consider nil-automorphisms of groups with residual properties namely locally-graded groups, residually-finite groups and profinite groups. The first result we prove says that a finite group of nil-automorphisms of a locally graded group, must be nilpotent. Next we turn our attention to groups of unipotent automorphisms of residually-finite and profinite groups. We show that such groups are locally-nilpotent and, as a by-product, we obtain an alternative proof of a well known theorem of Wilson about n-Engel residually-finite groups.