The minimal faithful permutation degree for a direct product obeying an inequality condition

The minimal faithful permutation degree $\mu(G)$ of a finite group $G$ is the least nonnegative integer $n$ such that $G$ embeds in the symmetric group \Sym(n). Clearly $\mu(G \times H) \le \mu(G) + \mu(H)$ for all finite groups $G$ and $H$. Wright (1975) proves that equality occurs when $G$ and $H$ are nilpotent and exhibits an example of strict inequality where $G\times H$ embeds in \Sym(15). Saunders (2010) produces an infinite family of examples of permutation groups $G$ and $H$ where $\mu(G \times H) < \mu(G) + \mu(H)$, including the example of Wright's as a special case. The smallest groups in Saunders' class embed in \Sym(10). In this paper we prove that 10 is minimal in the sense that $\mu(G \times H) = \mu(G) + \mu(H)$ for all groups $G$ and $H$ such that $\mu(G\times H)\le 9$.Comment: 22 page

Periodic elements of the free idempotent generated semigroup on a biordered set

We show that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup

Braids and factorizable inverse monoids

What is the untangling effect on a braid if one is allowed to snip a string, or if two specified strings are allowed to pass through each other, or even allowed to merge and part as newly reconstituted strings? To calculate the effects, one works in an appropriate factorizable inverse monoid, some aspects of a general theory of which are discussed in this paper. The coset monoid of a group arises, and turns out to have a universal property within a certain class of factorizable inverse monoids. This theory is dual to the classical construction of fundamental inverse semigroups from semilattices. In our braid examples, we will focus mainly on the ``merge and part'' alternative, and introduce a monoid which is a natural preimage of the largest factorizable inverse submonoid of the dual symmetric inverse monoid on a finite set, and prove that it embeds in the coset monoid of the braid group

Presentations of factorizable inverse monoids

It is well-known that an inverse monoid is factorizable if and only if it is a homomorphic image of a semidirect product of a semilattice (with identity) by a group. We use this structure to describe a presentation of an arbitrary factorizable inverse monoid in terms of presentations of its group of units and semilattice of idempotents, together with some other data. We apply this theory to quickly deduce a well known presentation of the symmetric inverse monoid on a nite set

On regularity and the word problem for free idempotent generated semigroups

The category of all idempotent generated semigroups with a prescribed structure E of their idempotents E (called the biordered set) has an initial object called the free idempotent generated semigroup over E, defined by a presentation over alphabet E, and denoted by IG(E). Recently, much effort has been put into investigating the structure of semigroups of the form IG(E), especially regarding their maximal subgroups. In this paper we take these investigations in a new direction by considering the word problem for IG(E). We prove two principal results, one positive and one negative. We show that, for a finite biordered set E, it is decidable whether a given word w ∈ E∗represents a regular element; if in addition one assumes that all maximal subgroups of IG(E) have decidable word problems, then the word problem in IG(E) restricted to regular words is decidable. On the other hand, we exhibit a biorder E arising from a finite idempotent semigroup S, such that the word problem for IG(E) is undecidable, even though all the maximal subgroups have decidable word problems. This is achieved by relating the word problem of IG(E) to the subgroup membership problem in finitely presented groups

Student Perspectives on Summer School Versus Term-Time for Undergraduate Mathematics

Earlier studies at The University of Sydney indicate that students undertaking certain first year mathematics units in intensive mode of delivery (IMD) achieved superior learning outcomes compared to those completing the same units during the semester. The aim of this study is to survey students that took any undergraduate mathematics units offered in IMD over the period 2009-2016, asking them to compare summer school with semester learning environments. While data suggest that the learning environment is overwhelmingly in favour of summer school, there are features of both modes that appear to be successful. This leads to a flow-diagram, akin to Biggs’ Presage-Process-Product (3P) model, emphasising presage and temporality

Are Labour Markets Necessarily Local? Spatiality, Segmentation and Scale

This paper draws on recent debates about scale to approach the geography of labour markets from a dynamic perspective sensitive to the spatiality and scale of labour market restructuring. Its exploration of labour market reconfigurations after the collapse of a major firm (Ansett Airlines) raises questions about geography’s faith in the inherently ‘local’ constitution of labour markets. Through an examination of the job reallocation process after redundancy, the paper suggests that multiple labour markets use and articulate scale in different ways. It argues that labour market rescaling processes are enacted at the critical moment of recruitment, where social networks, personal aspirations and employer preferences combine to shape workers’ destinations