Minimal classes and maximal class inp-groups

The number of conjugacy classes of a given size (not 1) in ap-group is divisible byp-1. We study groups in which the number of classes of minimal size is exactlyp-1, and characterise metabelian groups and groups of maximal class with this property. Part of the work of this author was done during his visit to the University of Napoli, under a CNR grant. He is grateful to that university for its hospitality, and also to Y. Berkovich for interesting discussions on the subject matter of this note.     

A simple group generated by involutions interchanging residue classes of the integers

We present a countable simple group which arises in a natural way from the arithmetical structure of the ring of integers.     

Divisor class group and canonical class of rings defined by ideals of pfaffians

In this paper we study the rings defined by ideals of pfaffians of a skew symmetric matrix of indeterminates. We analyze the case in which the pfaffians are not necessarily of fixed size. We prove that such rings are Cohen-Macaulay normal domains and we compute the divisor class group and the canonical class. It allows us to determine which of our rings are Gorenstein.     

Factorization in dedekind domains with finite class group

LetD be a Dedekind domain. It is well known thatD is then an atomic integral domain (that is to say, a domain in which each nonzero nonunit has a factorization as a product of irreducible elements). We study factorization properties of elements in Dedekind domains with finite class group. IfD has the property that any factorization of an elementα into irreducibles has the same length, thenD is called a half factorial domain (HFD, see [41]). IfD has the property that any factorization of an elementα into irreducibles has the same length modulor (for somer>1), thenD is called a congruence half factorial domain of orderr. In Section I we consider some general factorization properties of atomic integral domains as well as the interrelationship of the HFD and CHFD property in the Dedekind setting. In Section II we extend many of the results of [41], [42] and [36] concerning HFDs when the class group ofD is cyclic. Finally, in Section III we consider the CHFD property in detail and determine some basic properties of Dedekind CHFDs. IfG is any Abelian group andS any subset ofG−[0], then {G, S} is called a realizable pair if there exists a Dedekind domainD with class groupG such thatS is the set of nonprincipal classes ofG which contain prime ideals. We prove that for a finite abelian groupG there exists a realizable pair {G, S} such that any Dedekind domain associated to {G, S} is CHFD for somer>1 but not HFD if and only ifG is not isomorphic toZ ₂,Z ₂,Z ₂ ⊕Z ₂, orZ ₃ ⊕Z ₃. The first author received support under the John M. Bennett Fellowship at Trinity University and also gratefully acknowledges the support of The University of North Carolina at Chapel Hill.     

The fitting class generated by a finite soluble group

Sunto Per definizione una classe di Fitting di gruppi risolubili contiene i sottogruppi normali di ogni suo gruppo, e anche ogni gruppo il quale è prodotto di sottogruppi normali della classe. Un probleme assai difficile riguarda la classe di Fitting generata da un gruppo G finito risolubile: precisare proprietà strutturali in modo che H giace nella classe se e solo se gode di queste proprietà. Dimostriamo da un teorema di riduzione per gruppi G di struttura sufficientemente limitata. L'ispirazione è il lavoro di McCann (Examples of minimal Fitting Classes of finite groups, Arch. Math.,49 (1987), 179–186), e anche quello di Dark (Some examples in the theory of injectors of finite soluble groups, Math. Z.,127 (1972), 145–156). Inoltre costruiamo dei nuovi esempi di classi di Fitting.     

Topological algebras with maximal regular ideals closed

It is shown that all maximal regular ideals in a Hausdorff topological algebra A are closed if the von Neumann bornology of A has a pseudo-basis which consists of idempotent and completant absolutely pseudoconvex sets. Moreover, all ideals in a unital commutative sequentially Mackey complete Hausdorff topological algebra A with jointly continuous multiplication and bounded elements are closed if the von Neumann bornology of A is idempotently pseudoconvex.     

On the intersection of the maximal ideals

Some additional properties of the intersection of the maximal ideals of a compact semigroup are developed here based on results in [1] and [3]. Throughout S denotes a compact usually connected semigroup with at least one maximal proper ideal. The set of all such maximal ideals is denoted by ℳ, the intersection of the members of ℳ by R, the idempotents by E and the minimal ideal by K. Some proofs are more algebraic and in a few cases we do not need S connected. Key facts are that members of ℳ are open and dense, complements of distinct maximal ideals are disjoint and the union of any two such is S. After some generalization of results in [1] and [3], we investigate R relative to the topology of S. Necessary and sufficient conditions are found for R to be compact hence closed and for R to be open. Unlike the situation with the minimal ideal, R can be closed or open largely depending on the position of E relative to R. The following theorem summarizes necessary preliminaries from [1] and [3]. Adapted from material in Chapter 3 of the author's dissertation, written under the co-direction of Dr. John Mack and Dr. John Selden at the University of Kentucky and supported by the National Science Foundation. Support to organize and prepare this paper was provided by Mount Vernon Nazarene College.     

The normal fitting class generated by a finite soluble group

Sunto Per ogni gruppo finito risolubile G, la classe di Fitting normale generata da G può essere descritta mediante certe coppie di Fitting introdotte da Laue, Lausch e Pain in [4] e riprese da Berger in [2]. Si ottiene in particolare una caratterizzazione piuttosto semplice della classe di Fitting normale generata da S₃.     

The extended mapping class group is generated by 3 symmetries

We prove that for g?1 the extended mapping class group is generated by three orientation reversing involutions. To cite this article: M. Stukow, C. R. Acad. Sci. Paris, Ser. I 338 (2004).