# 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

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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#### References

[1] R. Baer, Engelsche Elemente Noetherscher Gruppen, Math. Ann., 133 (1957) 256–270.
[2] C. Casolo and O. Puglisi, Nil-automorphisms of groups with residual properties, Israel J. Math. 198 (2013) 91–110.
[3] G. Crosby and G. Traustason, On right n-Engel subgroups, J. Algebra, 324 (2010) 875–883.
[4] M. Frati, Unipotent automorphisms of soluble groups with finite Pr¨ufer rank, J. Group Theory, 17 (2014) 419–432.
[5] K.W. Gruenberg, The Engel elements of a soluble group, Illinois J. Math., 3 (1959) 151–168.
[6] G. Havas and M. R. Vaughan-Lee, 4-Engel groups are locally nilpotent, Internat. J. Algebra Comput., 15 (2005),
649–682.
[7] B. Hartley, Finite groups of automorphisms of locally finite soluble groups, J. Algebra, 57 (1979) 241–257.
[8] H. Heineken, Engelsche Elemente del L¨ange drei Gruppen, Illinois J. Math., 5 (1961) 681–707.
[9] P. Neumann, Pathology in the representation theory of infinite soluble groups, GroupsKorea 1988 (Pusan, 1988),
Lecture Notes in Math., 1398, Springer, Berlin, 1989 124139.
[10] Y. Medvedev, On compact Engel groups, Israel J. Math., 135 (2003) 147–156.
[11] B. I. Plotkin, Groups of automorphisms of algebraic systems, Wolters-Noordhoff Publishing, 1972.
[12] O. Puglisi and G. Traustason, Unipotent automorphisms of solvable groups, J. Group Theory, 20 (2017) 573–578.
[13] E. I. Zelmanov, Some problems in the theory of groups and Lie algebras, Math. Sb., 180 (1989) 159–167.
[14] J. S. Wilson and E. I. Zelmanov, Identities for Lie algebras of pro-p groups, J. Pure Appl. Algebra, 81 (1992)
103–109.