Solvable groups with cyclic relation module
Let \(G\) be a group, \(P\) be a group of operators on \(G\), \(d_ P(G)\) be the minimal number of generators of \(G\) as a \(P\)-group. Let \(G \cong F/N\) be a presentation of \(G\), where \(F\) is a free group. Then \(F\) acts on \(N\) by conjugation and induces an action of \(G\) on \(N_{ab} = N/[N,N]\). The \(\mathbb{Z} G\)-module \(N_{ab}\) is called the relation module of the presentation \(F/N\). A presentation \(G \cong F/N\) is an \(m\)-relator presentation if \(d_ F(N) = m\), it is an almost \(m\)-relator presentation if \(d_ G(N_{ab}) = m\). The main results are as follows: Theorem 1. Let \(F/N\) be an almost \(m\)-relator presentation of a finitely generated abelian group. Then this presentation is an \(m\)-relator presentation. Let \(G\) be a group with presentation \(G \cong F/N\). Then \(G\) is called an almost one-relator group if the relation module \(N_{ab}\) is a cyclic \(\mathbb{Z} G\)-module. Theorem 2. If \(G\) is a finitely generated soluble almost one-relator group, then \(G\) is either cyclic or \(G = \langle x,y\mid yxy^{-1} = x^ m\rangle\) for some \(0 \neq m \in \mathbb{Z}\). So the soluble almost one-relator groups are precisely the soluble one-relator groups. Theorem 8. Let \(G\) be a finitely generated soluble group, \(G \cong F/N\) be a presentation of \(G\) such that the \(\mathbb{Z} G\)-module \(N_{ab}\) is cyclic. Then \(N\) is the normal closure of a single element in \(F\).
finitely generated soluble almost one-relator group
scientific article; zbMATH DE number 500593