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

Coclass theory for nilpotent semigroups via their associated algebras

Coclass theory has been a highly successful approach towards the investigation and classification of finite nilpotent groups. Here we suggest a similar approach for finite nilpotent semigroups. This differs from the group theory setting in that we additionally use certain algebras associated to the considered semigroups. We propose a series of conjectures on our suggested approach. If these become theorems, then this would reduce the classification of nilpotent semigroups of a fixed coclass to a finite calculation. Our conjectures are supported by the classification of nilpotent semigroups of coclass 0 and 1. Computational experiments suggest that the conjectures also hold for the nilpotent semigroups of coclass 2 and 3.Comment: 13 pages, 2 figure

Consternation, confrontation and collaboration: narratives of medical students broaching obesity in primary care

This thesis explores one of the most important but neglected areas of medical education: obesity. I have worked in this area in an educational capacity with final year medical students for five years. I have published and presented at conferences extensively, and have become increasingly critical of the prevailing research paradigm in this area. I have had time to reflect on my journey and the drivers behind my educational practice, which were reflective narratives of my students’ first consultations with obese patients. I have identified and analysed the stories that informed change to my academic practice through a retroductive explanatory narrative analysis: why might this student, in this context, write this story in this way? Why was it impactful on my practice at that time, and in what ways did it inform change? Finally, I have theorised about how practitioner narratives create educational value for the narrator, researcher and reader

Recognising the small Ree groups in their natural representations

We present Las Vegas algorithms for constructive recognition and constructive membership testing of the Ree groups 2G_2(q) = Ree(q), where q = 3^{2m + 1} for some m > 0, in their natural representations of degree 7. The input is a generating set X. The constructive recognition algorithm is polynomial time given a discrete logarithm oracle. The constructive membership testing consists of a pre-processing step, that only needs to be executed once for a given X, and a main step. The latter is polynomial time, and the former is polynomial time given a discrete logarithm oracle. Implementations of the algorithms are available for the computer algebra system MAGMA

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

Success and limiting factors in health service innovation: a theory generating mixed-methods evaluation of UK projects

Objectives: To explore and explain success and limiting factors in UK health service innovation. Design: Mixed methods evaluation of a series of health service innovations involving a survey and interviews, with theory-generating analysis. Setting: The research explored innovations supported by one of the UK’s Academic Health Science Networks which provides small grants, awards and structural support to health service innovators including clinical academics, health and social care professionals and third-sector organisations. Participants: All recipients of funding or support 2014–2018 were invited to participate. We analysed survey responses relating to 56 innovation projects. Results: Responses were used to conceptualise success along two axes: value creation for the intended beneficiaries and expansion beyond its original pilot. An analysis of variance between categories of success indicated that participation, motivation and evaluation were critical to value generation; organisational, educational and administrative support were critical to expansion; and leadership and collaborative expertise were critical to both value creation and expansion. Additional limiting factors derived from qualitative responses included difficulties navigating the boundaries and intersections between organisations, professions, sectors and cultures; a lack of support for innovation beyond the start-up phase; a lack of protected time; and staff burn-out and turnover. Conclusions: A nested hierarchy of innovation needs has been derived via an analysis of these factors, providing targeted suggestions to enhance the success of future innovations. Data availability statement Data are available upon reasonable request. Due to the highly individual nature of healthcare innovations and the limited geographic area of this study, we are unable to provide our raw data. We undertake to provide a redacted data set upon reasonable request