Infinite groups with an Engel condition on infinite subsets (Q5939030)

[This review concerns also the preceding item Zbl 0981.20027.] The two articles under review treat problems of the following form: what can be said about a group \(G\) belonging to a class \(C\) in which every infinite subset contains a pair \((x,y)\) satisfying a condition \(B\)? A first statement in this direction was exhibited by B. H. Neumann for the class \(C\) of all groups and \([x,y]=1\) for \(B\). In the first paper the class \(C\) of locally graded groups is considered, the condition \(B\) is ``generating a \(k\)-Engel group''; these are shown to be extensions of finite groups by \(E_k\)-groups. In the second paper the class \(C\) consists of finitely generated residually finite groups, the condition \(B\) reads ``satisfies a two-Engel condition'', and here \(G/Z_2(G)\) is finite. Also, the condition ``satisfies an \(n\)-Engel condition'' is considered.