The 2A Majorana representations of the Harada-Norton group.

We show that all 2 A -Majorana representations of the Harada- Norton group F 5 have the same shape. If R is such a representation, we determine, using the theory of association schemes, the dimension and the irreducible constituents of the linear span U of the Majorana axes. Finally, we prove that, if R is based on the (unique) embedding of F 5 in the Monster, U is closed under the algebra product

Permutation modules for the symmetric group

In this paper we present a general method for computing the irreducible components of the permutation modules of the symmetric groups over a field $F$ of characteristic 0. We apply this machinery to determine the decomposition into irreducible submodules of the $F[S_n]$-permutation module on the right cosets of the normaliser in $S_n$ of the subgroup generated by a permutation of type $(3,3)$

Characterization and BMP Tests of Liquid Substrates for High-rate Anaerobic Digestion

This work was focused on the physicochemical characterization and biochemical methane potential (BMP) tests of some liquid organic substrates, to verify if they were suitable for undergoing a process of high-velocity anaerobic digestion. The selected substrates were: first and second cheese whey, organic fraction of municipal solid waste (OFMSW) leachate, condensate water and slaughterhouse liquid waste. Firstly, a physicochemical characterization was performed, using traditional and macromolecular parameters; then, batch anaerobic tests were carried out, and some continuous tests were planned. The results revealed that all the analyzed substrates have a potential to be anaerobically treated. Valuable information about treatment rate for a high-velocity anaerobic digestion process was obtained. Start-up of a lab-scale UASB reactor, treating diluted cheese whey, was successfully achieved with good COD removal efficiency. These preliminary results are expected to be further investigated in a successive phase, where continuous tests will be conducted on condensate water and OFMSW leachate. This work is licensed under a Creative Commons Attribution 4.0 International License

Feasibility assessment of reclaimed wastewater reuse in agriculture: how we do it

The growing interest towards wastewater (WW) reuse as alternative irrigation source is raised by the worldwide concern on water shortages and enhanced by the new European Directive on water reuse minimum requirements. In this perspective, water utilities and decision makers would benefit from a methodology to evaluate and encourage safe and efficient agricultural WW reuse practices. In this work, we propose a novel approach to identify criteria for assessing and prioritizing WW treatment plants (WWTPs) suitability for WW reuse practices implementation. The developed methodology, coupling WWTPs’ characteristics (i.e., flowrate and effluent quality) and features of the local territory (i.e., cultivated crops and climate), is able to quantify the economic savings, in terms of water and nutrients, and avoided environmental impacts, that could be fulfilled from WW reuse, and which WWTPs and territories to prioritize in its implementation

La falsificazione epigrafica. Questioni di metodo e casi di studio

This paper aims to reconsider the manuscript by Jacopo Valvasone (1499-1570), formerly owned by the Earl of Leicester (now British Library, Additional MS 49369), which Theodor Mommsen borrowed and inspected in 1876, just before the publication of the second part of CIL V. In the letter that he wrote to thank the Vicar and Librarian of Halkham Hall, Mommsen declared that Valvasone joined \u201cthe the long list of forgers\u201d. The analysis of forgeries in Valvasone\u2019s manuscript could show whether Mommsen was right in his opinion

A Simple Proof of Baer's and Sato's Theorems on Lattice Isomorphisms between Groups

A group $G$ is said to be an M\sp*-group if all subgroups of $G$ are quasinormal and $G$ is quaternionfree. Using Iwasawa's characterization of M\sp*-groups the author gives an elegant and unified proof of the following theorem: If $G$ is an M\sp*-group, then there exists an abelian group $A$ such that the lattices of subgroups of $A$ and $G$ are isomorphic. [J.Chvalina (Brno)

Modular sublattices in a finite p-group

Der Verf. studiert den Durchschnitt aller maximalen modularen Teilverb\ue4nde des Untergruppenverbandes {\goth L}(G) einer Gruppe $G$ und nennt ihn {\goth Z}(G), da er genau die von Zacher bereits 1955 studierten Untergruppen enth\ue4lt, die mit jedem modularen Teilverband einen modularen Teilverband von {\goth L}(G) erzeugen. Nat\ufcrlich geh\uf6ren 1 und $G$ zu {\goth Z}(G), und der Verf. zeigt, dass eine der folgenden drei Aussagen gilt, wenn $G$ eine endliche $p$-Gruppe mit $p>2$ und M\in{\goth Z}(G) mit $1<M<G$ ist:\par (1) {\goth L}(G) ist modular.\par (2) $M\le\langle x^p\mid x\in H\rangle$ f\ufcr jedes $H\le G$ mit {\goth L}(H) nicht modular.\par (3) $M$ enth\ue4lt jede nicht permutable Untergruppe von $G$.\par Der Verf. gibt Beispiele f\ufcr $p$-Gruppen mit Elementen von {\goth Z}(G), die Eigenschaft (2) bzw. (3) haben. Ob aber jedes solche $M$ zu {\goth Z}(G) geh\uf6rt, bleibt offen. [R.Schmidt (Kiel)

Projektivitaeten zwischen abelschen und nichtabelschen Gruppen

In 1951 S. Sato showed that the lattice of subgroups of a modular group with elements of infinite order is isomorphic to the one of a convenient abelian group. Recently in the last part of Sato's work some inexactitudes were found which could question the validity of the result. \par In this paper a new proof of that theorem is provided. Included are also two results on modular groups with elements of infinite order. Namely that any such group can be embedded in a modular group whose torsion- subgroup is divisible and that in case the torsion-subgroup is divisible a modular group splits into the semidirect product of its torsion- subgroup by a cyclic, or locally cyclic, group

Gruppi con pochi sottogruppi non quasinormali

Let $G$ be a group. A subgroup $H$ of $G$ is said to be quasinormal if $HK = KH$ for every subgroup $K$ of $G$, and $G$ is called quasi-Hamiltonian if all its subgroups are quasinormal. The structure of quasi-Hamiltonian groups has been described by {\it K. Iwasawa} [in J. Fac. Sci., Univ. Tokyo, Sect. I 4, 171-199 (1941; Zbl 0061.025) and Jap. J. Math. 18, 709-728 (1943; Zbl 0061.025)]. If $G$ is a group, let $Q(G)$ denote the subgroup generated by all subgroups of $G$ which are not quasinormal. Then $G$ is quasi-Hamiltonian if and only if $Q(G) = 1$, and it is easy to show that $Q(G)$ is generated by all cyclic non-quasinormal subgroups of $G$.\par The author studies the class $\bold X$ of all groups $G$ for which $Q(G)$ is a proper subgroup. The corresponding problem for the subgroup generated by all non-normal subgroups was considered by {\it D. Cappitt} [J. Algebra 17, 310-316 (1971; Zbl 0232.20067)]. Clearly every $\bold X$-group is generated by cyclic quasinormal subgroups, and in particular it is locally nilpotent. The author proves that non-periodic $\bold X$-groups are quasi-Hamiltonian. The investigation of periodic $\bold X$-groups can be reduced to the case of a $p$-group ($p$ prime), and the description of $p$-groups in the class $\bold X$ is obtained. In particular, it is shown that a $p$-group of infinite exponent $G$ is in the class $\bold X$ if and only if the subgroup generated by all non-normal subgroups of $G$ is properly contained in $G$. Finally, the author proves that if $G$ is an $\bold X$-group whose Sylow 2-subgroup is quasi-Hamiltonian, then $G$ is metabelian. [F.de Giovanni (Napoli)