Subnormal closure of a homomorphism

Let $\varphi\colon\Gamma\to G$ be a homomorphism of groups. In this paper we introduce the notion of a subnormal map (the inclusion of a subnormal subgroup into a group being a basic prototype). We then consider factorizations $\Gamma\xrightarrow{\psi} M\xrightarrow{n} G$ of $\varphi,$ with $n$ a subnormal map. We search for a universal such factorization. When $\Gamma$ and $G$ are finite we show that such universal factorization exists: $\Gamma\to\Gamma_{\infty}\to G,$ where $\Gamma_{\infty}$ is a hypercentral extension of the subnormal closure $\mathcal{C}$ of $\varphi(\Gamma)$ in $G$ (i.e.~the kernel of the extension $\Gamma_{\infty}\to {\mathcal C}$ is contained in the hypercenter of $\Gamma_{\infty}$). This is closely related to the a relative version of the Bousfield-Kan $\mathbb{Z}$-completion tower of a space. The group $\Gamma_{\infty}$ is the inverse limit of the normal closures tower of $\varphi$ introduced by us in a recent paper. We prove several stability and finiteness properties of the tower and its inverse limit $\Gamma_{\infty}$.Comment: 13 pages

Normal and conormal maps in homotopy theory

Let M be a monoidal category endowed with a distinguished class of weak equivalences and with appropriately compatible classifying bundles for monoids and comonoids. We define and study homotopy-invariant notions of normality for maps of monoids and of conormality for maps of comonoids in M. These notions generalize both principal bundles and crossed modules and are preserved by nice enough monoidal functors, such as the normaliized chain complex functor. We provide several explicit classes of examples of homotopy-normal and of homotopy-conormal maps, when M is the category of simplicial sets or the category of chain complexes over a commutative ring.Comment: 32 pages. The definition of twisting structure in Appendix B has been reformulated, leading to further slight modifications of definitions in Section 1. To appear in HH

The first record of Merycomyia whitneyi (Johnson), tribe Bouvieromyiini (Diptera: Tabanidae), from Texas and from west of the Mississippi River

The first collections of Merycomyia whitneyi (Johnson), (Diptera: Tabanidae: Chrysopsinae: Bouvieromyiini) from Texas and from west of the Mississippi River are reported, and the Nearctic species of the Tribe Bouvieromyiini are discussed

The flavor of product-group GUTs

The doublet-triplet splitting problem can be simply solved in product-group GUT models, using a global symmetry that distinguishes the doublets from the triplets. Apart from giving the required mass hierarchy, this ``triplet symmetry'' can also forbid some of the triplet couplings to matter. We point out that, since this symmetry is typically generation-dependent, it gives rise to non-trivial flavor structure. Furthermore, because flavor symmetries cannot be exact, the triplet-matter couplings are not forbidden then but only suppressed. We construct models in which the triplet symmetry gives acceptable proton decay rate and fermion masses. In some of the models, the prediction m_b ~ m_\tau is retained, while the similar relation for the first generation is corrected. Finally, all this can be accomplished with triplets somewhat below the GUT scale, supplying the right correction for the standard model gauge couplings to unify precisely.Comment: 10 page