Metabelian Groups and Related Topics

Abstract

The title of this chapter does not reflect its main purpose. We shall discuss metabelian groups here and not under the heading of “special groups” in Chapter 11.10, where they would seem to belong, because their first investigation was motivated by problems arising in three different fields outside of group theory. Within a few years, Furtwängler [1930], Reidemeister [1932a], and Moufang [1937] used metabelian groups to prove theorems pertaining to algebraic number theory, knot theory, and the foundations of geometry, respectively. We shall give some background information about these problems, starting with a few remarks about the authors, and, after discussing the particular theorems on metabelian groups involved in the solution of these problems, we shall give a brief account of some of the later developments arising from or related to the original investigations.