Group Theory

The Microfoundations of Community: Small Groups As Bridges and Barriers to Participatory Democracy

by Gazi Islam

Notes on Groups and Geometry, 1978-1986, by Steven H. Cullinane

by Steven H. Cullinane

Typewritten notes.

The Diamond Theorem

by Steven H. Cullinane

Expanded version of a result originally published as Abstract 79T-A37, "Symmetry invariance in a diamond ring," in the Notices of the American Mathematical Society, February 1979, pp. A193-A194. The result also appeared in an earlier preprint, "Diamond Theory," distributed in 1976.

Finite projective geometry explains the surprising symmetry properties of some simple graphic designs-- found, for... more Finite projective geometry explains the surprising symmetry properties of some simple graphic designs-- found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.

Immunity of information encoded in decoherence-free subspaces to particle loss

by Piotr Migdal

Using group theory to catalyse productivity and creativity within a band scenario

by Jake Whiteley

Copyright: Jake Whiteley, 2011
Un-published

This essay serves as a case study into the group dynamics present within a typical band scenario. It attempts to... more This essay serves as a case study into the group dynamics present within a typical band scenario. It attempts to utilise various group theories through the action - reflection method and stimulate productivity and creativity within a small group.

Comparing single and joint preferences: a choice experiment on residential location in three-member households

by marcucci edoardo

Residential location preferences and relative power in three-member households- a choice experiment

by marcucci edoardo

REAL REPRESENTATIONS OF FINITE GROUPS

by Stefan Karpinski

Infinite primitive directed graphs

by Simon Smith

Appeared in the J. Algebraic Combin., (2010)

doi: 10.1007/s10801-009-0190-3

A graph X has connectivity one if it is connected and there exists a vertex the removal of which leaves X... more A graph X has connectivity one if it is connected and there exists a vertex the removal of which leaves X disconnected. If X has connectivity one, a lobe of X is a connected subgraph that is maximal subject to the condition that it does not have connectivity one.

The primitive undirected graphs with connectivity one have been fully classified by Jung and Watkins: the lobes of such graphs are primitive, pairwise-isomorphic and have at least three vertices. When one considers the general case of a directed primitive graph with connectivity one, however, this result no longer holds. In this paper we investigate the structure of these directed graphs, and obtain a complete characterisation.

Orbital graphs of infinite primitive permutation groups

by Simon Smith

Appeared in J. Group Theory, (2007)

doi: 10.1515/JGT.2007.060

If G is a group acting on a set V and a, b are elements of V, the digraph whose vertex set is V and whose arc set is... more If G is a group acting on a set V and a, b are elements of V, the digraph whose vertex set is V and whose arc set is the orbit (a, b)^G is called an orbital digraph of G.

A locally finite digraph X has more than one end if there exists a finite set of vertices Y such that the induced digraph X \ Y contains at least two infinite connected components; if there exists such a set containing precisely one element, then X has connectivity one.

In this paper we show that if G is a primitive permutation group whose suborbits are all finite, possessing an orbital digraph with more than one end, then G has a primitive connectivity-one orbital digraph, and this digraph is essentially unique. Such digraphs resemble trees in many respects, and have been fully characterised in another paper (Infinite primitive digraphs) by the author.

Subdegree growth rates of infinite primitive permutation groups

by Simon Smith

Appeared in the J. London Math. Soc., (2010)

doi: 10.1112/jlms/jdq046

A transitive group G of permutations of a set V is primitive if the only G-invariant equivalence relations on V are... more A transitive group G of permutations of a set V is primitive if the only G-invariant equivalence relations on V are the trivial and universal relations. The orbits of a point-stabiliser acting on V are called the suborbits of G and the cardinalities of these suborbits are the subdegrees of G.

If G acts primitively on an infinite set V, and all the suborbits of G are finite, Adeleke and Neumann asked if, after enumerating the subdegrees of G as a non-decreasing sequence, the subdegree growth rates of infinite primitive groups that act distance-transitively on locally finite distance-transitive graphs are extremal, and conjecture that it may be possible to determine the group from the rate of growth.

In this paper it is shown that such an enumeration is not desirable. The examples used to show this provide several novel methods for constructing infinite primitive graphs.
A revised enumeration method is then proposed, and it is shown that, under this, Adeleke and Neumann's question may be answered, at least for groups exhibiting suitable rates of growth.

Rough ends of infinite primitive groups

by Simon Smith

Appeared in the J. Group Theory, (2011)

doi: 10.1515/JGT.2011.108

If G is a group of permutations of a set V, then the suborbits of G are the orbits of point-stabilisers acting on V.... more If G is a group of permutations of a set V, then the suborbits of G are the orbits of point-stabilisers acting on V. The cardinalities of these suborbits are the subdegrees of G. Every infinite primitive permutation group G with finite subdegrees acts faithfully as a group of automorphisms of a locally-finite connected vertex-primitive directed graph X with vertex set V, and there is consequently a natural action of G on the ends of X.

We show that if G is closed in the permutation topology of pointwise convergence, then the structure of G is determined by the length of any orbit of G acting on the ends of X.

Distinguishability of infinite groups and graphs

by Simon Smith

Preprint, co-authored with Tom Tucker and Mark Watkins

The distinguishing number of a group G acting faithfully on a set V is the least number of colors needed to color the... more The distinguishing number of a group G acting faithfully on a set V is the least number of colors needed to color the elements of V so that no non-identity element of the group preserves the colouring. The distinguishing number of a graph is the distinguishing number of its full automorphism group acting on its vertex set.

We prove that every connected primitive graph with infinite diameter and countably many vertices has distinguishing number 2. Consequently, any infinite, connected, primitive, locally finite graph is 2-distinguishable; so, too, is any infinite primitive group with finite suborbits. We also show that all denumerable vertex-transitive graphs of connectivity 1 and all Cartesian products of connected denumerable graphs of infinite diameter have distinguishing number 2. All of our results follow directly from a versatile lemma which we call The Distinct Spheres Lemma.

A classification of primitive permutation groups with finite stabilizers

by Simon Smith

Preprint

We classify all infinite primitive permutation groups possessing a finite point stabilizer, thus extending the seminal... more We classify all infinite primitive permutation groups possessing a finite point stabilizer, thus extending the seminal O'Nan--Scott Theorem to all primitive permutation groups with finite point stabilizers.

Irreducible Representations of Baumslag-Solitar Groups

by Daniel McLaury

We classify the finite-dimensional irreducible linear representations of the Baumslag-Solitar groups BS(p,q) = < a,... more We classify the finite-dimensional irreducible linear representations of the Baumslag-Solitar groups BS(p,q) = < a, b | a b^p = b^q a > for relatively prime p and q. The general strategy of the argument is to consider the matrix group given by image of a representation and study its Zariski closure in GL(n, C).

Group Cohesiveness: The Smoothest Way to Goal Achievement

by Musab Wahedna

Flashgangs and Flashgangbanging: How can local police prepare?

by Carter Smith

Keywords: flash gang, flash gangbanging, flash mobs, flash robs, gangs

Flash mobs, participants in an event in which a group of people are organized via some form of telecommunications, assemble suddenly in a public place, perform an unusual and sometimes seemingly pointless act for a brief time, and then disperse. A recent phenomenon synthesizes the activity of flash mobs and street gangs. Communities everywhere have experienced the negative effects of street gangs, and their proliferation has led to an increase in destructive crimes in the United States. A flash gang is a group that uses a social media connection to invite participants to a time and location where they commit a crime and then they split up. Local law enforcement needs to examine response policies to prepare for this new and dangerous phenomenon. Recent protests in the Middle East and North Africa were coordinated using similar strategies, spotlighting the power of using social media technology to oppose government action. The spontaneity and secrecy of the flash mob combined with the targeted crime and/or violence of the street gang produces a mix that is hard to combat even with inside intelligence. The instant access and extended reach of mobile phones and social media sites like Twitter and Facebook bring a twist that makes the spontaneous volatility even more difficult to prevent.

The response thus far by law enforcement seems to be the monitoring of social media (Facebook, Twitter, etc.). This may be the best and only way to know what’s going on in these instances. Of course the questions remain whether this will ultimately be constitutional and what do, how does law enforcement respond with this information that they have. Does this call for a dedicated team like a digital SWAT Team? Is the problem that police are not trained to respond to groups? Police are generally equipped to respond to individual for interaction, arrest, and prosecution, but throughout the criminal justice system there has been little training regarding organizational behavior.

Flash mobs, participants in an event in which a group of people are organized via some form of telecommunications,... more Flash mobs, participants in an event in which a group of people are organized via some form of telecommunications, assemble suddenly in a public place, perform an unusual and sometimes seemingly pointless act for a brief time, and then disperse. A recent phenomenon synthesizes the activity of flash mobs and street gangs. Communities everywhere have experienced the negative effects of street gangs, and their proliferation has led to an increase in destructive crimes in the United States. A flash gang is a group that uses a social media connection to invite participants to a time and location where they commit a crime and then they split up. Local law enforcement needs to examine response policies to prepare for this new and dangerous phenomenon. Recent protests in the Middle East and North Africa were coordinated using similar strategies, spotlighting the power of using social media technology to oppose government action. The spontaneity and secrecy of the flash mob combined with the targeted crime and/or violence of the street gang produces a mix that is hard to combat even with inside intelligence. The instant access and extended reach of mobile phones and social media sites like Twitter and Facebook bring a twist that makes the spontaneous volatility even more difficult to prevent.

"Who Needs Yalom When We Have Žižek?"

by William J. Urban

paper published in the International Journal of Zizek Studies, v.2, n.2 (2008)

Rehearsing popular music: Exploring opportunities for supporting learning in the pop/rock band

by Mark Pulman

There seems little reported about group-based rehearsals of popular music and the peer learning opportunities that... more There seems little reported about group-based rehearsals of popular music and the peer learning opportunities that might arise from this activity. Although there are an increasing number of studies exploring approaches to the assessment of musical ensembles, these often focus on performance rather than rehearsing and, typically, do not specifically address popular music courses (Hunter, 2006). Indeed, Lebler (2008) describes popular music as being usually learned in the broader community as a self-directed activity, sometimes including interactions with peers and group activities, but rarely under the direction of an expert mentor/teacher. The role of the tutor, in facilitating learning opportunities that may be available for students working in popular music genres within a band rehearsal context, can be quite different to that required for rehearsing repertoire which might be described as being drawn from western art music traditions. If so, and given the apparent lack of literature on, and pedagogical resources for, band rehearsing of popular music (Lebler, 2007) within the HE curriculum, the aim of this project is to provide a contribution towards filling that gap.

Dynamic Stereotype Representations: Roles and Scenarios that Impact Groups' Interaction

by Vlad Glaveanu

Published by The International Journal of Diversity in Organisations, Communities and Nations, 2007, 7(3), 65-76

The present paper discusses a new concept: dynamic stereotype representations, a line of research in the field of... more The present paper discusses a new concept: dynamic stereotype representations, a line of research in the field of social cognition and attitudes inspired by the Transactional Analysis conception regarding interpersonal relationships (mainly the States of the Ego theory). The main assumption is that, besides stereotypes, specific stereotype-roles are associated with groups or social categories. These roles have a major impact upon inter-group relationships since groups tend to fit into a stereotype-scenario that “predicts” the outcome of these relationships and by this acts like a prophecy that transforms the social reality. The objective of the paper is to address the issue of dynamic stereotype representations Romanians have about themselves and about European Union citizens in the context of Romania becoming an EU member in January 2007. The survey was based on a samples of 105 Romanians (two age groups: 64 students and 41 teachers), investigating stereotypes (using a 12 attributes scale validated in previous research), stereotype-roles (using a special validated questionnaire) and resulting stereotype-scenarios. Our hypotheses assume that: 1) the stereotype image attributed to EU citizens will be more favourable than the one of their own national group; 2) the social creativity strategy will make Romanians value themselves more on the “Agreeableness” dimension while EU citizens will be valued more on the “Competence” dimension; 3) we can identify stereotype-roles attributed both to Romanians and EU citizens and these are different in our two age-groups; 4) the stereotype-roles associated with Romanians represent Child figures while the ones attributed to EU citizens are especially Mature type of roles; 5) the identified stereotype-roles will help us depict a stereotype-scenario that reflects the inter-group relationship as anticipated by our Romanian participants. Even if the core nature of this research is exploratory the results confirmed our assumptions and led us to new findings and their practical implications.

Keywords: Stereotype, Stereotype-Role, Stereotype-Scenario, Transactional Analysis, Romanians and EU Citizens