24,822 research outputs found
Groups whose prime graphs have no triangles
Let G be a finite group and let cd(G) be the set of all complex irreducible
character degrees of G Let \rho(G) be the set of all primes which divide some
character degree of G. The prime graph \Delta(G) attached to G is a graph whose
vertex set is \rho(G) and there is an edge between two distinct primes u and v
if and only if the product uv divides some character degree of G. In this
paper, we show that if G is a finite group whose prime graph \Delta(G) has no
triangles, then \Delta(G) has at most 5 vertices. We also obtain a
classification of all finite graphs with 5 vertices and having no triangles
which can occur as prime graphs of some finite groups. Finally, we show that
the prime graph of a finite group can never be a cycle nor a tree with at least
5 vertices.Comment: 13 page
Groups with normal restriction property
Let G be a finite group. A subgroup M of G is said to be an NR-subgroup if,
whenever K is normal in M, then K^G\cap M=K, where K^G is the normal closure of
K in G. Using the Classification of Finite Simple Groups, we prove that if
every maximal subgroup of G is an NR -subgroup then G is solvable. This gives a
positive answer to a conjecture posed in Berkovich (Houston J Math 24:631-638,
1998).Comment: 5 page
Large-eddy simulation for flow and dispersion in urban streets
Large-eddy simulations (LES) with our recently developed inflow approach (Xie &Castro, 2008a) have been used for flow and dispersion within a genuine city area -the DAPPLE site, located at the intersection of Marylebone Rd and Gloucester Plin Central London. Numerical results up to second-order statistics are reported fora computational domain of 1.2km (streamwise) x 0.8km (lateral) x 0.2km (in fullscale), with a resolution down to approximately one meter in space and one secondin time. They are in reasonable agreement with the experimental data. Such a comprehensiveurban geometry is often, as here, composed of staggered, aligned, squarearrays of blocks with non-uniform height and non-uniform base, street canyons andintersections. Both the integrative and local effect of flow and dispersion to thesegeometrical patterns were investigated. For example, it was found that the peaksof spatially averaged urms, vrms, wrms and < u0w0 > occurred neither at the meanheight nor at the maximum height, but at the height of large and tall buildings. Itwas also found that the mean and fluctuating concentrations in the near-source fieldis highly dependent on the source location and the local geometry pattern, whereasin the far field (e.g. >0.1km) they are not. In summary, it is demonstrated thatfull-scale resolution of around one meter is sufficient to yield accurate prediction ofthe flow and mean dispersion characteristics and to provide reasonable estimationof concentration fluctuation
Character degree sums in finite nonsolvable groups
Let N be a minimal normal nonabelian subgroup of a finite group G. We will
show that there exists a nontrivial irreducible character of N of degree at
least 5 which is extendible to G. This result will be used to settle two open
questions raised by Berkovich and Mann, and Berkovich and Zhmud'.Comment: 5 page
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