Thomas A. Fournelle

Automorphisms of nilpotent groups of class two with small rank

Rational abelian groups, that is, torsion-free abelian groups of rank one, are characterized by their types. This paper characterizes rational nilpotent groups of class two, that is, nilpotent groups of class two in which the center and central factor group are direct sums of rational abelian groups. This characterization is done according to the types of the summands of the center and the central factor group. Using these types and some cohomological techniques it is possible to determine the automorphism group of the nilpotent group in question by performing essentially matrix computations. In particular, the automorphism groups of rational nilpotent groups of class two and rank three are completely described. Specific examples are given of semicomplete and pseudocomplete nilpotent groups.

Properties that characterize classes of nilpotent groups

ConclusionObviously we have not covered aU possible ascendant and descendant nilpotent properties. Using (1.1), (1.2), or other methods, and Lemma 1.2 it is possible to provide many more such properties. For exampleCq∼(Cpm×Cpm) is a metabelian group having infinitely many non-isomorphic Sylow-p-subgroups [6]. Thus, having one isomorphism class of Sylow-p-subgroups is a descendant nilpotent property.The importance of such properties lies not only in their utility within the class of nilpotent groups but in the fact that they delineate, in the sense of (1.3) and (1.4), the class of nilpotent groups within the class of solvable groups.