Frank B. Cannonito

The algorithmic theory of polycyclic-by-finite groups☆

A group is said to be polycyclic-by-finite, or a PF-group for short, if it has a polycyclic normal subgroup of finite index. Equivalently, PF-groups are exactly the groups which have a series of finite length whose infinite factors are cyclic. By a well-known theorem of P. Hall every PF-group is finitely presented-and in fact PF-groups form the largest known sectionclosed class of finitely presented groups. It is this fact that makes PF-groups natural objects of study from the algorithmic standpoint. The general aim of the algorithmic theory of PF-groups can be described as the collection of information about a PF-group which can, in principle

Computable algebra and group embeddings

This work grew out of our investigations of the infinitely generated subgroups of finitely presented groups. A useful technique for embedding countable groups with sufficiently nice local properties into finitely presented groups was introduced in [l]. This technique was then applied in [ 1, 21 to prove that every countable locally polycyclic group and every countable metabelian group can be embedded in a finitely presented group. It became apparent from [2] that effective methods in commutative algebra could be extended to group rings of polycyclic groups and then used to embed certain groups in finitely presented groups. In this paper these extensions are carried out with the consequent application that every countable locally abelian-by-nilpotent group can be embedded in a finitely presented group. Much of the effective commutative algebra we require seems to be known. (See for instance [6, 9, 14, 171.) Unfortunately a systematic account is not available. In particular we need a proof of the Hilbert basis theorem which

On Primitive Recursive Permutations and their Inverses

Abstract : It is known that there is a primitive recursive permutation of the nonnegative integers whose inverse is recursive but not primitive recursive. Robinson showed that every singulary recursive function f is representable as f = A(B superscript -1)C, where A, B, C are primitive recursive and B is a permutation. This report presents a sharper version of Robinsons result, that is, every singulary recursive f is of the form f = A(B superscript -1)C for fixed A,C where A,B,C are elementary functions and B is a permutation. The proof employs meta-mathematical methods. (Author)

A note on normal and nonnormal subgroups of given index

A well-known elementary proposition of group theory of interest in algebraic coding theory asserts that any subgroup of index 2 must be normal. Less well known is its generalization: i fp is the smallest prime dividing the order of the group, any subgroup of index p is normal. In this note we give an elementary proof of the generalization and, by way of contrast, show for each integer n > 2 how to consm~ct a group Gn with nonnormal subgroup lln of index n. First, the generalization.

A Note on Inverses of Elementary Permutations

It is shown that there exists an elementary permutation of the natural numbers whose inverse is not provably recursive in formal number theory. This result is interpreted with respect to choosing the provably recursive functions as the appropriate formalization of the class of functions effectively computable on intuitive grounds.