Generalized Engel elements

Generalized Engel elements in Group Theory, J. Group Theory 6 No. 1 (2003) 69-81.

In the theory of nilpotent groups, we have the notions of right Engel elements and left Engel elements, defined in terms of repeating commutators. The hypercentre and the Hirsch-Plotkin radical are subsets of the sets of right and left Engel elements respectively.

Towards the end of my PhD, we were considering analogues of Engel elements but for supersoluble conditions. We define left and right S-Engel elements. There is a supersoluble analogue of the hypercentre, whose members are right S-Engel elements and we define an analogue of the Hirsch-Plotkin radical, whose members are left S-Engel elements. These analogue subgroups are precisely the relevant S-Engel elements in certain cases (such as a polycyclic-by-finite group).

This paper was based on work in my PhD thesis conducted under the supervision of Prof B.A.F. Wehrfritz and with the support of an EPSRC grant.

Abstract

This paper provides a super soluble analogue of Engel theory. We show that the theory works for finite groups and list some further results pertaining to finitary skew linear groups.

Dedicated to Bert Wehrfritz on his 60th birthday.

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