2 - Mal'cev's theorems

Summary

In this chapter we obtain some general results of a qualitative nature about the structure of polycyclic groups, at a somewhat deeper level than those of Chapter 1, by beginning to exploit the ‘linear’ aspect of these groups. At its simplest, this comes down to the observation that if A/B is a free abelian factor, of rank n say, in a group G (with B < A both normal subgroups of G), then the action of G by conjugation on this factor affords a representation of G in Aut(A/B) = GL_n(ℤ). In later chapters we probe more deeply into the precise nature of the link between polycyclic groups and linear groups over ℤ.

Rationally irreducible modules

In the study of a finite soluble group G, a fruitful line of attack is to pick a minimal normal subgroup N ≠ 1 of G and investigate the action of G on N. If we try the same thing when G is infinite and polycyclic, we find that usually there is no such N; if N does exist, it is always finite and so makes an insignificant contribution to the structure of G (for example we have h(G/N) = h(G)). What we do, instead, is to consider a free abelian normal subgroup A≠1 of G of minimal rank. A then has the structure of a rationally irreducible G/A-module, and about such things much can be said.