Algorithmic Methods for Finitely Generated Abelian Groups

Finitely Generated Abelian Groups

MATH 436 Notes: Finitely generated Abelian groups

Definition 1.1 (Direct Products). Let {Gα}α∈I be a collection of groups indexed by an index set I. We may form the Cartesian product ∏ α∈I Gα. The elements of this Cartesian product can be denoted by tuples (aα)α∈I . We refer to the entry aα as the αth component of this tuple. We define a multiplication on this Cartesian product componentwise, i.e., (aα) ⋆ (bα) = (aα ⋆α bα) where ⋆α is the grou...

The Structure Theorem for Finitely Generated Abelian Groups

This paper provides a thorough explication of the Structure Theorem for Abelian groups and of the background information necessary to prove it. The outline of this paper is as follows. We first consider some theorems related to abelian groups and to R-modules. In this section we see that every finitely generated abelian group is the epimorphic image of a finitely generated free abelian group. H...

Which Finitely Generated Abelian Groups Admit Equal Growth Functions?

We show that finitely generated Abelian groups admit equal growth functions with respect to symmetric generating sets if and only if they have the same rank and the torsion parts have the same parity. In contrast, finitely generated Abelian groups admit equal growth functions with respect to monoid generating sets if and only if they have same rank. Moreover, we show that the size of the torsio...

The Algorithmic Theory of Finitely Generated Metabelian Groups

Algorithms are constructed which, when an explicit presentation of a finitely generated metabelian group G in the variety si1 is given, produce unitary presentations for the derived subgroup G', the centre Z{G), the Fitting subgroup Fit(G), and the Frattini subgroup <p(G). Additional algorithms of independent interest are developed for commutative algebra which construct the associated set of p...