On torsion in finitely presented groups

We give a uniform construction that, on input of a recursive presentation $P$ of a group, outputs a recursive presentation of a torsion-free group, isomorphic to $P$ whenever $P$ is itself torsion-free. We use this to re-obtain a known result, the existence of a universal finitely presented torsion-free group; one into which all finitely presented torsion-free groups embed. We apply our techniques to show that recognising embeddability of finitely presented groups is $\Pi^{0}_{2}$-hard, $\Sigma^{0}_{2}$-hard, and lies in $\Sigma^{0}_{3}$. We also show that the sets of orders of torsion elements of finitely presented groups are precisely the $\Sigma^{0}_{2}$ sets which are closed under taking factors.Comment: 11 pages. This is the version submitted for publicatio

Coset intersection graphs for groups

Let H, K be subgroups of G. We investigate the intersection properties of left and right cosets of these subgroups.Comment: 4 page

Torsion, torsion length and finitely presented groups

Abstract We show that a construction by Aanderaa and Cohen used in their proof of the Higman Embedding Theorem preserves torsion length. We give a new construction showing that every finitely presented group is the quotient of some C ′ ⁢ ( 1 / 6 ) {C^{\prime}(1/6)} finitely presented group by the subgroup generated by its torsion elements. We use these results to show there is a finitely presented group with infinite torsion length which is C ′ ⁢ ( 1 / 6 ) {C^{\prime}(1/6)} , and thus word-hyperbolic and virtually torsion-free.</jats:p

Finitely annihilated groups

We say a group is finitely annihilated if it is the set-theoretic union of all its proper normal finite index subgroups. We investigate this new property, and observe that it is independent of several other well known group properties. For finitely generated groups, we show that in many cases it is equivalent to having non-cyclic abelianisation, and at the same time construct an explicit infinite family of counterexamples to this. We show for finitely presented groups that this property is neither Markov nor co-Markov. In the context of our work we show that the weight of a non-perfect finite group, or a non-perfect finitely generated solvable group, is the same as the weight of its abelianisation. We generalise a theorem of Brodie-Chamberlain-Kappe on finite coverings of groups, and finish with some generalisations and variations of our new definition.Comment: 13 pages. This is the version submitted for publicatio

Mathematical Artifacts Have Politics: The Journey from Examples to Embedded Ethics

We extend Langdon Winner's idea that artifacts have politics into the realm of mathematics. To do so, we first provide a list of examples showing the existence of mathematical artifacts that have politics. In the second step, we provide an argument that shows that all mathematical artifacts have politics. We conclude by showing the implications for embedding ethics into mathematical curricula. We show how acknowledging that mathematical artifacts have politics can help mathematicians design better exercises for their mathematics students

A Hippocratic Oath for mathematicians? Mapping the landscape of ethics in mathematics

While the consequences of mathematically-based software, algorithms and strategies have become ever wider and better appreciated, ethical reflection on mathematics has remained primitive. We review the somewhat disconnected suggestions of commentators in recent decades with a view to piecing together a coherent approach to ethics in mathematics. Calls for a Hippocratic Oath for mathematicians are examined and it is concluded that while lessons can be learned from the medical profession, the relation of mathematicians to those affected by their work is significantly different. There is something to be learned also from the codes of conduct of cognate but professionalised quantitative disciplines such as engineering and accountancy, as well as from legal principles bearing on professional work. We conclude with recommendations that professional societies in mathematics should sponsor an (international) code of ethics, institutional mission statements for mathematicians and syllabuses of ethics courses for incorporation into mathematics degrees

The subgroup identification problem for finitely presented groups

We introduce the subgroup identification problem, and show that there is a finitely presented group G for which it is unsolvable, and that it is uniformly solvable in the class of finitely presented locally Hopfian groups. This is done as an investigation into the difference between strong and weak effective coherence for finitely presented groups.Comment: 11 pages. This is the version submitted for publicatio