Determination of a certain family of finite metabelian groups

Abstract

Introduction. The problem of constructing all the finite metabelian groups (that is, groups with an abelian commutator subgroup) is fundamentally settled by the Schreier theory of group extensions. In fact, a metabelian group may be considered as the extension of an abelian group 2a by an abelian group a; this is the simplest nontrivial case of a group extension. According to Schreier's theory, an arbitrary group 5 which is the extension of 2 by a is obtained by the following procedure: First, we have to find an automorphism group of 2f which is the homomorphic image of a, that is, if ais the automorphism corresponding to a Ea, then (AB) =A Ba, (A)a =Aa 5=A for every A, B GX, o, rEG. Secondly, we have to find a factor system Ca, in A, satisfying CaT, CaT= C7,P for every pi o, ir (see [18, p. 90])(l). If Sa is a symbol denoting a certain representative of oin 0, then the relations SaAS,i =Aa, S,S, = C?,7Sa7 uniquely determine an abstract group 5 with the required properties. Unfortunately, the general formulation of the Schreier theory does not indicate (except in the most trivial cases) how to determine and specify the automorphisms and factor systems so that each 5 shall be obtained in one and only one way. The invariant characterization of solvable groups, even in the relatively simple case of metabelian groups, still remains one of the most important and most difficult problems of abstract group theory. At the present, it seems that the problem can be successfully approached only if we impose certain restrictions upon the family of groups to be determined. In a recent paper [12] I have determined all the groups (5 which have an abelian invariant subgroup 21 of the type (p, * *, p) such that 9/2 be cyclic. In the present paper a more extensive class of metabelian groups will be determined and completely characterized by numerical invariants. Whereas no restriction will be imposed upon the structure of the abelian invariant subgroup X, it is assumed that 9/f is cyclic and its order n is not divisible by the square of any prime number which divides the order of 2W. In particular, the latter condition is fulfilled if either n is squarefree, or if n and the order of 2a are relatively prime. Among the more important cases included in the above category (others will be mentioned in part 3) perhaps the most notable is the case of p-groups