An l-group G is an abelian group equipped with a translation invariant lattice order. Baker and Beynon proved that G is finitely generated projective iff it is finitely presented. A unital l-group is an l-group G with a distinguished order unit, i.e., an element 0 ≤ u ∈ G whose positive integer multiples eventually dominate every element of G. While every finitely generated projective unital l-...

We show that any finitely generated metabelian group can be embedded in a metabelian group of type F3. More generally, we prove that if n is a positive integer and Q is a finitely generated abelian group, then any finitely generated ZQ-module can be embedded in a module that is n-tame. Combining with standard facts, the F3 embedding theorem follows from this and a recent theorem of R. Bieri and...

We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a finitely presented abelian-by-nilpotent group by its largest nilpotent normal subgroup.

In this paper we use basic concepts of Quantum Computing to try to solve the problem of finding generators of a finitely generated abelian group. We present three algorithms based on random selection of group elements to find a minimal generating set for a finite abelian group. We also describe an algorithm for finding a minimal generating set for a torsion free abelian group.

3 Groups 12 3.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Subgroups, Cosets and Lagrange’s Theorem . . . . . . . . . . . . . . . . . . 14 3.3 Finitely Generated Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4 Permutation Groups and Group Actions . . . . . . . . . . . . . . . . . . . . 19 3.5 The Oribit-Stabiliser Theorem . ....

The girth of a finitely generated group G is defined to be the supremum of the girth of its Cayley graphs. Let G be a finitely generated subgroup of the mapping class group ModΣ, where Σ is an orientable closed surface with a finite number of punctures and with a finite number of components. We show that G is either a non-cyclic group with infinite girth or a virtually free-abelian group; these...

This paper surveys the methods that have been used to attack conjecture, still open, an abelian variety over a characteristic $0$ field with finitely generated Galois group is always of infinite rank.