Counterexamples to a conjecture of Lemmermeyer

We produce infinitely many finite 2-groups that do not embed with index 2 in any group generated by involutions. This disproves a conjecture of Lemmermeyer and restricts the possible Galois groups of unramified 2-extensions, Galois over the rationals, of quadratic number fields

The Largest Condorcet Domains on 8 Alternatives

In this note, we report on a record-breaking Condorcet domain (CD) for n=8 alternatives. We show that there exists a CD of size 224, which is optimal and essentially unique (up to isomorphism). If we consider the underlying permutations and focus on Condorcet domains containing the identity permutation, 56 isomorphic such Condorcet domains exist. Our work sheds light on the structure of CDs and UCDs and has potential applications in voting theory and social choice

Condorcet Domains of Degree at most Seven

In this paper we give the first explicit enumeration of all maximal Condorcet domains on $n\leq 7$ alternatives. This has been accomplished by developing a new algorithm for constructing Condorcet domains, and an implementation of that algorithm which has been run on a supercomputer. We follow this up by the first survey of the properties of all maximal Condorcet domains up to degree 7, with respect to many properties studied in the social sciences and mathematical literature. We resolve several open questions posed by other authors, both by examples from our data and theorems. We give a new set of results on the symmetry properties of Condorcet domains which unify earlier works. Finally we discuss connections to other domain types such as non-dictatorial domains and generalisations of single-peaked domains. All our data is made freely available for other researches via a new website.Comment: The paper is still undergoing some revision

The matrix group recognition project; where do we go from here?

Non UBCUnreviewedAuthor affiliation: Queen Mary College LondonResearche