On right \(n\)-Engel subgroups. II.
This interesting and important paper is a continuation of part I [J. Algebra 324, No. 4, 875-883 (2010; Zbl 1205.20045)] by the same authors. The main object of the paper are ``right \(n\)-Engel subgroups'': A subgroup \(H\) of a group \(G\) is called a right \(n\)-Engel subgroup if all elements of \(H\) are right \(n\)-Engel elements of \(G\). And to speak about the main problem, recall that as a ``Global Application'' of Zel'manov's celebrated solution of the Restricted Burnside Problem, there exists a function \(f\colon\mathbb N\times\mathbb N\to\mathbb N\) such that every nilpotent \(n\)-Engel \(d\)-generator group \(G\) has nilpotency class at most \(f(n,d)\). The latter means \(G\leq Z_{f(n,d)}(G)\) or \([G,{_{f(n,d)}G}]=1\). A natural question is now as follows: If one only assumes that an ``appropriate part'' \(H\) of the group \(G\) has an ``\(n\)-Engel property'', is it possible to find a function \(g\) similar to \(f\) such that \(H\subseteq Z_{g(n,d)}(G)\) or, what is the same thing, \([H,{_{g(n,d)}G}]=1\); or to find some ``similar/near'' consequences of the latter. In part I [loc. cit.], the authors proved some expected (not at all trivial) results by proving a ``Local Application'' of Zel'manov's results on Lie rings. For example, in Theorem 3 of part I the authors answered Question 3.81 of \textit{A. Abdollahi}, [``Engel elements in groups'', in Groups St.~Andrews 2009. Vol. I. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 387, 94-117 (2011)], which is the same as Problem 7.12 of [\textit{V.D. Mazurov} (ed.) and \textit{E.I. Khukro} (ed.), The Kourovka notebook. Unsolved problems in group theory. 17th ed. Novosibirsk: Institute of Mathematics, Russian Academy of Sciences, Siberian Div. (2010; Zbl 1211.20001)]. In this article, the authors prove the ``local'' analogues of Propositions 2.3 and 2.3 and Theorem 2.4 of \textit{A. Abdollahi} and \textit{G. Traustason} [Proc. Am. Math. Soc. 130, No. 10, 2827-2836 (2002; Zbl 1006.20029)] as Theorems 5, 6 and 7. Theorem 4 is proved by using Lemma 3 of part I, some commutator calculus and specifically by an interesting application of the Hall-Petrescu identity. In Section 3, the authors find the least upper bound for the functions given in their Theorems 4, 5, 6 and 7 whenever the subgroup is right \(2\)-Engel.
scientific article; zbMATH DE number 5888508