SOLVABLE-GROUPS WITH CYCLIC RELATION MODULE

[摘要] If P is a group of operators on a group A, then we denote by d(P)(A) the minimal number of generators of A as a P-group. Suppose G is a group and F/N is a presentation of G. Then F acts on N by conjugation and induces an action of G on N(ab). This ZG-module is called the relation module of the presentation F/N. A presentation 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. It is an open problem whether every almost m-relator presentation is an m-relator presentation. This paper is part of a program to investigate this question when m = 1. We show that an almost one-relator presentation F/N is a one-relator presentation in case the group G = F/N is finitely generated and solvable.

[发布日期] 1993-12-01