Normal series of soluble groups of finite rank (Q1063114)

The main results of this paper are proved for the class of Mal'cev groups, that is, groups which have finite normal series with factors which are either finite abelian or aperiodic (torsion-free) abelian of finite rank. However, all the illustrative examples in the paper are polycyclic groups (groups which have finite series with cyclic factors), so the results will be stated here in simpler forms referring to the smaller class of polycyclic groups. A finite normal series of a polycyclic group G is called a normal constitution series (elsewhere called a weak chief series) if each factor is either a G-simple finite abelian group or a free abelian group with no G-subgroup of finite index, and the series is called strong if no normal constitution series of G has fewer finite factors. The greatest among the ranks of the free abelian factors of a normal constitution series of G is called the species of G. Let \(\rho\) (G) be the maximal finite normal subgroup of the polycyclic group G and let M/N be a finite factor in a strong normal constitution series of G. The factor M/N is called radical if there exists a non- trivial element of \(\rho\) (G) in M but not in N, otherwise M/N is non- radical. The main theorem of the paper is that the order of a non-radical factor of a polycyclic group of species n is a divisor of \(p^ m\), where p is any of the primes less than \(n+2\) and m is the integer part of n/(p-1). This theorem means that, although \(\rho\) (G) can be any finite soluble group, the structure of G/\(\rho\) (G) is controlled by the species n. An immediate consequence of the main theorem is that, for any polycyclic group G, the strong normal constitution factors of p-power order are invariants of G if \(p>n+1\), whereas this was known not to hold if \(p\leq n+1\). It also follows immediately that the Sylow p-subgroups of G are conjugate if \(p>n+1\), but it is shown that there exist polycyclic groups of species p-1 with Sylow p-subgroups of orders p and \(p^ 2\) for all odd primes p, a result that had already been obtained for 2 by G. Zappa. There is a difficulty in the use of the main theorem that the definition of a strong normal constitution series of a polycyclic group G involves a study of all such series, but this difficulty is resolved in this paper by means of a finite construction for the strong normal constitution series of G.