On a problem of P. Hall for Engel words

Abstract

Let θ be a word in n variables and let G be a group; the marginal and verbal subgroups of G determined by θ are denoted by θ(G) and θ*(G), respectively. The following problems are generally attributed to P. Hall: (I)If π is a set of primes and {pipe}G: θ*(G){pipe} is a finite π-group, is θ(G) also a finite π-group? (II)If θ(G) is finite and G satisfies maximal condition on its subgroups, is {pipe}G: θ*(G){pipe} finite? (III)If the set {θ(g1,...,gn) {pipe} g1,...,gn ∈ G} is finite, does it follow that θ(G) is finite? We investigate the case in which θ is the n-Engel word en = [x,n y] for n∈ {2,3,4}. © 2011 Springer Basel AG.