A characterization of nilpotent-by-finite groups in the class of finitely generated soluble groups
Titel:
A characterization of nilpotent-by-finite groups in the class of finitely generated soluble groups
Auteur:
Endimioni, G.
Verschenen in:
Communications in algebra
Paginering:
Jaargang 25 (1997) nr. 4 pagina's 1159-1168
Jaar:
1997
Inhoud:
Let a,b be elements of a group G. Suppose that there exists a relation of the form [image omitted] with m, n ∈ N[image omitted] and [image omitted] . Such relations generalize well-known relations as [image omitted] . Denote by Π* (a, b) (respectively Π * (a, b)) the least integer [image omitted] (respectively [image omitted] ) with this property and put Π* (a, b) = Π* (a, b) = +∞ if a and b do not satisfy a relation of the previous form. We prove that a finitely generated soluble group G is polycyclic if and only if for any a,b ∈ G, we have Π* (a, b) = 1. Moreover, a polycyclic group G is nilpotent-by-finite if and only if for any a,b ∈ G, the sequence Π* (a, bk)) k>0 is bounded. It follows that a finitely generated soluble group G is nilpotent-by-finite if and only if for any a,b ∈ G, Π* (a, b) = 1 and the sequence (Π* (a, bk)) k>0 is bounded.
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