Primitive permutation groups and derangements of prime power order

Let $G$ be a transitive permutation group on a finite set of size at least $2$. By a well known theorem of Fein, Kantor and Schacher, $G$ contains a derangement of prime power order. In this paper, we study the finite primitive permutation groups with the extremal property that the order of every derangement is an $r$-power, for some fixed prime $r$. First we show that these groups are either almost simple or affine, and we determine all the almost simple groups with this property. We also prove that an affine group $G$ has this property if and only if every two-point stabilizer is an $r$-group. Here the structure of $G$ has been extensively studied in work of Guralnick and Wiegand on the multiplicative structure of Galois field extensions, and in later work of Fleischmann, Lempken and Tiep on $r'$-semiregular pairs.Comment: 30 pages; to appear in Manuscripta Mat

Derangements in primitive permutation groups, with an application to character theory

Let $G$ be a finite primitive permutation group and let $\kappa(G)$ be the number of conjugacy classes of derangements in $G$. By a classical theorem of Jordan, $\kappa(G) \geqslant 1$. In this paper we classify the groups $G$ with $\kappa(G)=1$, and we use this to obtain new results on the structure of finite groups with an irreducible complex character that vanishes on a unique conjugacy class. We also obtain detailed structural information on the groups with $\kappa(G)=2$, including a complete classification for almost simple groups.Comment: 29 page

A Model of Quark and Lepton Masses I: The Neutrino Sector

If neutrinos have masses, why are they so tiny? Are these masses of the Dirac type or of the Majorana type? We are already familiar with the mechanism of how to obtain a tiny Majorana neutrino mass by the famous see-saw mechanism. The question is: Can one build a model in which a tiny Dirac neutrino mass arises in a more or less "natural" way? What would be the phenomenological consequences of such a scenario, other than just merely reproducing the neutrino mass patterns for the oscillation data? In this article, a systematic and detailed analysis of a model is presented, with, as key components, the introduction of a family symmetry as well as a new SU(2) symmetry for the right-handed neutrinos. In particular, in addition to the calculations of light neutrino Dirac masses, interesting phenomenological implications of the model will be presented.Comment: 25 (single-spaced) pages, 11 figures, corrected some typos in Table I, added acknowledgement

Interacting Business Cycle Fluctuations: A Two-Country Model

In this paper, we develop a model of business cycle fluctuations between two interacting open economies within the disequilibrium or non-market clearing paradigm. We analyze the main feedback mechanisms (Keynes, Mundell, Rose and Dornbusch) driving the dynamics and the conflict between their stabilizing and destabilizing tendencies and how these depend on certain key speeds of adjustment in the real and foreign exchange sectors. We explore numerically a variety of situations of interacting price cycles in the two countries, where the steady state is locally repelling, but where the overall dynamics are bounded in an economically meaningful domain by assuming downward money wage rigidity

Approximate gauge symmetry of composite vector bosons

It can be shown in a solvable field theory model that the couplings of the composite vector bosons made of a fermion pair approach the gauge couplings in the limit of strong binding. Although this phenomenon may appear accidental and special to the vector boson made of a fermion pair, we extend it to the case of bosons being constituents and find that the same phenomenon occurs in more an intriguing way. The functional formalism not only facilitates computation but also provides us with a better insight into the generating mechanism of approximate gauge symmetry, in particular, how the strong binding and global current conservation conspire to generate such an approximate symmetry. Remarks are made on its possible relevance or irrelevance to electroweak and higher symmetries.Comment: Correction of typos. The published versio