Elsa Fisman

Generalized table algebras

A table algebra was defined in [1] in order to consider in a uniform way the common properties of conjugacy classes and irreducible characters. Non-commutative table algebras were introduced in [5]. They generalize properties of such well-known objects as coherent and Hecke algebras. Here we extend the main definition of a non-commutative table algebra by letting the ground field be an integral domain. We call these algebrasgeneralized table algebras (GT-algebras, in brief). It is worth mentioning that this class of algebras includes generic Hecke-Iwahori algebras of finite Coxeter groups. We develop the general theory for this type of algebras which includes their representation theory and theory of closed subsets. We also study the properties of primitive integral table algebras.

On Finite Factorizable Groups

In this paper we use freely the recent classification theorem for finite simple groups. From the early stages of finite group theory, finite factorizable groups satisfying (H) were studied by many authors. In 1904 Burnside [6] proved the solvability of groups factorizable by two prime power subgroups. Many years later Wielandt [29] proved the solvability of finite groups factorizable by two nilpotent subgroups of relatively prime orders (Kegel extended Wielandt’s result in the case that both subgroups A and B are nilpotent without restrictions). The second author in [ 121 classified finite factorizable groups of type (H) where A and B are solvable subgroups. Results on groups satisfying (H) can be found in [3, 4, 10, 12, 19, 20, 291 and in other papers on finite factorizable groups. Theorem 1.1 and Corollary 1.2 of this paper state the final solution to the problem of classifying groups of type (H). We prove the following:

An analogy between products of two conjugacy classes and products of two irreducible characters in finite groups

It is well-known that the number of irreducible characters of a finite group G is equal to the number of conjugate classes of G . The purpose of this article is to give some analogous properties between these basic concepts.

About products of irreducible characters and products of conjugacy classes in finite groups

Recently, the research on products of ordinary irreducible characters and products of conjugacy classes in finite groups developed considerably. In many articles analogous theorems for these two kinds of products were proved EACH, ACM, AF, AH, AHS, AL, B, BC, Ch, Ma]. For example, it was proved in [ACM] that a finite group

Integral Table Algebras with a Faithful Nonreal Element of Degree 4

Throughout this chapter we assume that (A,B) is a standard integral table algebras generated by a non-real element of degree 4 and min(B) ≥ 3. The structure of the algebras from the class A strongly depends on the structure of the multiset \( [b\bar b] \) where b is a non-real faithful element of B of degree 4. A direct verification shows that the following multisets exhibit all possibilities for \( [b\bar b] \) : [14,62],[14,43],[14,34],[14,42,41], [14,41,41,41],[14,81,41],[14,121].

Standard Integral Table Algebras with a Faithful Nonreal Element of Degree 5

This chapter deals with the classification of standard integral GT-algebras (A,B) with L(B) = 1 {1} and |b| ≥ 4 for all b ∈ B# which contain a nonreal faithful basis element b of degree 5. Starting from this point using the basic identity

A proof of Szep's conjecture on nonsimplicity of certain finite groups

\lambda _{xyz} |z|\left\langle {xy,z} \right\rangle = \left\langle {x,z\bar y} \right\rangle = \lambda _{z\bar yx} |x|,x,y,z \in B,

Table algebras and applications to products of characters in finite groups

one can list all possible representations of \( b\bar b \) and b 2 as linear combinations of basis elements (cf. Tables II and III of Subsection 3.3). Assuming that b commutes with \( \bar b \) yields the identity \( \left\langle {b\bar b,b\bar b} \right\rangle = \left\langle {b^2 ,b^2 } \right\rangle \) which reduces the number of these representations (cf. Table III of Subsection 3.3). Then, using various kind of techniques (for example repeated application of the associa- tivity law), each of the reamining cases will be treated separately. In order to state the main result, we introduce the following base of a specific table algebra.