Some remarks on unipotent automorphisms

Document Type: Proceedings of the conference "Engel conditions in groups" - Bath - UK - 2019

Authors

1 Dipartimento di Matematica, viale Morgagni 67A

2 University of Bath

10.22108/ijgt.2020.119749.1581

Abstract

An automorphism $\alpha$ of the group $G$ is said to be $n$-unipotent if $[g,_n\alpha]=1$ for all $g\in G$‎. ‎In this paper we obtain some results related to nilpotency of groups of $n$-unipotent automorphisms of solvable groups‎. ‎We also show that‎, ‎assuming the truth of a conjecture about the representation theory of solvable groups raised by P‎. ‎Neumann‎, ‎it is possible to produce‎, ‎for a suitable prime $p$‎, ‎an example of a f.g‎. ‎solvable group possessing a group of $p$-unipotent automorphisms which is isomorphic to an infinite Burnside group‎. ‎Conversely we show that‎, ‎if there exists a f.g‎. ‎solvable group $G$ with a non nilpotent $p$-group $H$ of $n$-automorphisms‎, ‎then there is such a counterexample where $n$ is a prime power and $H$ has finite exponent‎.

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