A Construction of Metabelian Groups

In 1934, Garrett Birkhoff has shown that the number of isomorphism classes of finite metabelian groups of order p 22 tends to infinity with p. More precisely, for each prime number p there is a family (M λ) λ=0,... ,p−1 of indecom-posable and pairwise nonisomorphic metabelian p-groups of the given order. In this manuscript we use recent results on the classification of possible embeddings of a subgroup in a finite abelian p-group to construct families of indecom-posable metabelian groups, indexed by several parameters, which have upper bounds on the exponents of the center and the commutator subgroup. A group G is called metabelian if its commutator subgroup C 1 (G) is contained in the center C(G); it follows that the subgroup embedding C 1 (G) ⊆ C(G) is an isomorphism invariant of the group G. In [1, §15], Birkhoff observes that each such invariant can be realized, and he uses his construction of pairwise nonisomorphic subgroup embeddings to obtain a family, indexed by one parameter λ = 0,. .. , p−1, of pairwise nonisomorphic metabelian p-groups. We recall his result in Theorem 2. In fact, the classification of all subgroup embeddings is a problem considered infeasible. It is shown in [3] that for n > 6 the category S(Z/p n) of all embeddings of a subgroup in a p n-bounded finite abelian group is of (controlled) wild representation type and hence admits families of indecomposable and pairwise nonisomorphic objects which depend on several parameters. We obtain corresponding statements about families of metabelian groups in Theorem 3 and in Corollary 4. In these examples and in Birkhoff's, the exponent of the commutator subgroup is p 4. We show in Theorem 5 that this exponent can be reduced to p 3 , at the expense of a higher order and a larger exponent of the group. Birkhoff's Construction of Metabelian Groups Let B be a finite abelian p n-bounded group and A a subgroup of B where the embedding is denoted by ι : A → B. Define the finite group M = G(A ⊆ B) as the semidirect product M = (A ⊕ B) × ψ D where D = Z/p m is the cyclic group of order equal to the exponent of A, and ψ : D → Aut(A ⊕ B) is the group map given by ψ(d)(a, b) = (a, b + dι(a)). Note that additive notation is used …