The product of all the irreducible characters of a finite simple group

The content of this paper corresponds to a part of the author's Ph.D. thesis carried out in Tel Aviv University under the supervision of Prof. Marcel Herzog.     

Equlibrium measure of a product subset of

In this note we show that an equilibrium measure of a product of two subsets of and , respectively, is a product of their equilibrium measures. We also obtain a formula for , where are locally bounded plurisubharmonic functions and .

     

Equations on the semidirect product of a finite semilattice by a finite commutative monoid

LetCom _t,q denote the variety of finite monoids that satisfy the equationsxy=yx andx ^t =x ^t+q . The varietyCom _1,1 is the variety of finite semilattices also denoted byJ ₁. In this paper, we consider the product varietyJ ₁*Com _t,q generated by all semidirect products of the formM*N withM∈J ₁ andN∈Com _t,q . We give a complete sequence of equations forJ ₁*Com _t,q implying complete sequences of equations forJ ₁*(Com∩A),J ₁*(Com∩G) andJ ₁*Com, whereCom denotes the variety of finite commutative monoids,A the variety of finite aperiodic monoids andG the variety of finite groups. This material is based upon work supported by the National Science Foundation under Grants No. CCR-9101800 and CCR-9300738. Many thanks to the referee for his valuable comments and suggestions.     

Realizing a fusion system by a single finite group

We show that every saturated fusion system can be realized as a full subcategory of the fusion system of a finite group. The result suggests the definition of an ‘exoticity index’ and raises some other questions which we discuss.     

FC-Groups whose central factor group can be embedded in a direct product of finite groups

We consider the question of whether an FC-group G in which the derived subgroup [G, G] is a subgroup of a direct product of finite groups must have its central factor group G/Z(G) also embeddable in a direct product of finite groups.     

Minimal covers of finite sets

We enumerate the minimal covers of a finite set S, classifying such covers by their cardinality, and also by the number of elements in S which they cover uniquely.     

Minimal solvable algebraic groups with a finite center

Translated from Matematicheskie Zametki, Vol. 50, No. 2, pp. 120–124, August, 1991.     

The permutation class group of a finite group

Analogously to the projective class group, the permutation class group of a finite group π can be defined as the group of equivalence classes of direct summands of integral permutation modules modulo permutation modules. It is shown that this group behaves nicely with respect to localization and completion, which then is used to prove that contrary to the projective class group - it is not always a torsion group. More precisely, the rank of the permutation class of group is computed.     

Residually finite dimensional and AF-embeddable -algebras

We show that every separable nuclear residually finite dimensional -algebras satisfying the Universal Coefficient Theorem can be embedded into a unital separable simple AF-algebra.

     

Minimal convex functions bounded below by the duality product

It is well known that the Fitzpatrick function of a maximal monotone operator is minimal in the class of convex functions bounded below by the duality product. Our main result establishes that, in the setting of reflexive Banach spaces, the converse also holds; that is, every such minimal function is the Fitzpatrick function of some maximal monotone operator. Whether this converse also holds in a nonreflexive Banach space remains an open problem.

     

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.     

The finite vertex-primitive and vertex-biprimitive

A complete classification is given for finite vertex-primitive and vertex-biprimitive -transitive graphs for . The classification involves the construction of new 4-transitive graphs, namely a graph of valency 14 admitting the Monster simple group , and an infinite family of graphs of valency 5 admitting projective symplectic groups with prime and (mod 8). As a corollary of this classification, a conjecture of Biggs and Hoare (1983) is proved.

     

Minimal non-quaternion-free finite 2-groups

We classify minimal non-quaternion-free finite 2-groups which possess a nonmodular proper subgroup.     

Minimal sets in finite rings

We describe how to calculate the (, )-minimal sets in any finite ring.