Permutation groups, simple groups and sieve methods

We show that the number of integers n ≤ x which occur as indices of subgroups of nonabelian finite simple groups, excluding that of An-1 in An, is ∼ hx/log x, for some given constant h. This might be regarded as a noncommutative analogue of the Prime Number Theorem (which counts indices n ≤ x of subgroups of abelian simple groups). We conclude that for most positive integers n, the only quasiprimitive permutation groups of degree n are Sn and An in their natural action. This extends a similar result for primitive permutation groups obtained by Cameron, Neumann and Teague in 1982. Our proof combines group-theoretic and number-theoretic methods. In particular, we use the classification of finite simple groups, and we also apply sieve methods to estimate the size of some interesting sets of primes

Glucagonoma-induced acute heart failure

Neuroendocrine tumours (NETs) represent a broad spectrum of tumours, of which the serotonin-producing carcinoid is the most common and has been shown to cause right ventricular heart failure. However, an association between heart failure and NETs other than carcinoid has not been established so far. In this case report, we describe a 51-year-old patient with a glucagon-producing NET of the pancreas who developed acute heart failure and even cardiogenic shock despite therapy. Heart failure eventually regressed after initialising i.v. treatment with the somatostatin analogue octreotide. Chromogranin A as a tumour marker was shown to be significantly elevated, and it decreased with clinical improvement of the patient. The effects of long-time stimulation of glucagon on the myocardium have not been studied yet; however, sarcoplasmic reticulum calcium leak can be discussed as a possible mechanism for glucagon-induced heart failure

Rough ends of infinite primitive permutation groups

If G is a group of permutations of a set Omega , then the suborbits of G are the orbits of point-stabilisers G_\alpha acting on Omega. 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 Gamma with vertex set Omega, and there is consequently a natural action of G on the ends of Gamma. 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 Gamma. Examining the ends of a Cayley graph of a finitely generated group to determine the structure of the group is often fruitful. B. Krön and R. G. Möller have recently generalised the Cayley graph to what they call a rough Cayley graph, and they call the ends of this graph the rough ends of the group. It transpires that the ends of Gamma are the rough ends of G, and so our result is equivalent to saying that the structure of a closed primitive group G whose subdegrees are all finite is determined by the length of any orbit of G on its rough ends

Tournaments and Even Graphs are Equinumerous

A graph is called \emph{odd} if there is an orientation of its edges and an automorphism that reverses the sense of an odd number of its edges, and \emph{even} otherwise. Pontus von Br\"omssen (n\'e Andersson) showed that the existence of such an automorphism is independent of the orientation, and considered the question of counting pairwise non-isomorphic even graphs. Based on computational evidence, he made the rather surprising conjecture that the number of pairwise non-isomorphic \emph{even graphs} on $n$ vertices is equal to the number of pairwise non-isomorphic \emph{tournaments} on $n$ vertices. We prove this conjecture using a counting argument with several applications of the Cauchy-Frobenius Theorem

Diagonally Neighbour Transitive Codes and Frequency Permutation Arrays

Constant composition codes have been proposed as suitable coding schemes to solve the narrow band and impulse noise problems associated with powerline communication. In particular, a certain class of constant composition codes called frequency permutation arrays have been suggested as ideal, in some sense, for these purposes. In this paper we characterise a family of neighbour transitive codes in Hamming graphs in which frequency permutation arrays play a central rode. We also classify all the permutation codes generated by groups in this family

Abnormal expression of p27kip1 protein in levator ani muscle of aging women with pelvic floor disorders – a relationship to the cellular differentiation and degeneration

BACKGROUND: Pelvic floor disorders affect almost 50% of aging women. An important role in the pelvic floor support belongs to the levator ani muscle. The p27/kip1 (p27) protein, multifunctional cyclin-dependent kinase inhibitor, shows changing expression in differentiating skeletal muscle cells during development, and relatively high levels of p27 RNA were detected in the normal human skeletal muscles. METHODS: Biopsy samples of levator ani muscle were obtained from 22 symptomatic patients with stress urinary incontinence, pelvic organ prolapse, and overlaps (age range 38–74), and nine asymptomatic women (age 31–49). Cryostat sections were investigated for p27 protein expression and type I (slow twitch) and type II (fast twitch) fibers. RESULTS: All fibers exhibited strong plasma membrane (and nuclear) p27 protein expression. cytoplasmic p27 expression was virtually absent in asymptomatic women. In perimenopausal symptomatic patients (ages 38–55), muscle fibers showed hypertrophy and moderate cytoplasmic p27 staining accompanied by diminution of type II fibers. Older symptomatic patients (ages 57–74) showed cytoplasmic p27 overexpression accompanied by shrinking, cytoplasmic vacuolization and fragmentation of muscle cells. The plasma membrane and cytoplasmic p27 expression was not unique to the muscle cells. Under certain circumstances, it was also detected in other cell types (epithelium of ectocervix and luteal cells). CONCLUSIONS: This is the first report on the unusual (plasma membrane and cytoplasmic) expression of p27 protein in normal and abnormal human striated muscle cells in vivo. Our data indicate that pelvic floor disorders are in perimenopausal patients associated with an appearance of moderate cytoplasmic p27 expression, accompanying hypertrophy and transition of type II into type I fibers. The patients in advanced postmenopause show shrinking and fragmentation of muscle fibers associated with strong cytoplasmic p27 expression