Zvi Arad

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.

New Criteria for the Solvability of Finite Groups

(see [6] or [ 7, p. 291]). It has been conjectured that a group is solvable if and only if it has a Hall 2’-subgroup and a Hall 3’subgroup (see [ 11). We verify this conjecture for groups whose composition factors are known simple groups. Along the way we also verify (for groups whose composition factors are known simple groups) a conjecture of Hall’s that if a group has a Hall (

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:

A criterion for the existence of normal π-complements in finite groups

Abstract The purpose of this article is to prove, using the classification of the finite simple groups, the following conjecture: Let π be a set of odd primes, then a finite group is π-homogeneous if and only if it is π′-closed. Using this, several open problems can be settled, including an affirmative answer to the following problem of Baer: Let G be a finite group and π ⊆ π ( G ). Suppose that G is both π-homogeneous and π′-homogeneous. Is G a direct product of a π-group and a π′-group? Finally, we note that the proof of the conjecture yields proof to some theorems proved earlier without using the classification of the finite simple groups.

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.

Powers of Characters of Finite Groups

LetG be a finite group and θ a complex character ofG. Define Irr(θ) to be the set of all irreducible constituents of θ andIrr(G) to be the set of all irreducible characters ofG. Thecharacter-covering number of a finite groupG, ccn(G), is defined as the smallest positive integer m such thatIrr(χm) =Irr(G) for allχ∈Irr(G)—{1G}. If no such positive integer exists we say that the character-covering-number ofG is infinite. In this article we show that a finite nontrivial groupG has a finite character-covering-number if and only ifG is simple and non-abelian and ifG is a nonabelian simple group thenccn(G) ⩽ k2 − 3k + 4, wherek is the number of conjugacy classes ofG. Then we show (using the classification of the finite simple groups) that the only finite group with a character-covering-number equal to two is the smallest Jankos group,J1. These results are analogous to results obtained previously concerning the covering of groups by powers of conjugacy classes. Other related results are shown.

Classification of Finite Groups with a CC-Subgroup

Abstract A proper subgroup M of a group G is called a CC-subgroup of G if the centralizer C G (m) of every m ∈ M # = M ∖ {1} is contained in M. In this paper we classify all finite groups containing a CC-subgroup, extending work of many authors.

On Even Generalized Table Algebras

Generalized table algebras were introduced in Arad, Fisman and Muzychuk (Israel J. Math.114 (1999), 29–60) as an axiomatic closure of some algebraic properties of the Bose-Mesner algebras of association schemes. In this note we show that if all non-trivial degrees of a generalized integral table algebra are even, then the number of real basic elements of the algebra is bounded from below (Theorem 2.2). As a consequence we obtain some interesting facts about association schemes the non-trivial valencies of which are even. For example, we proved that if all non-identical relations of an association scheme have the same valency which is even, then the scheme is symmetric.

A classification of 3CC-groups and applications to Glauberman-Goldschmidt theorem

Following Higman [I 51 a finite group of order divisible by 3 in which centralizers of 3-elements are 3-groups will be called a COO-group. A finite group with an &-subgroup M containing the centralizer of each of its nonidentity elements will be called a 3CC-group. Several authors have studied such groups. In [5, lo] there is a complete description of COO-groups whose &-subgroups are of order 3 and 9. Other approaches to the problem of classifying COO-groups and 3CC groups can be found in [6-11, 14, 15, 171 (this is a partial list). Fletcher’s Corollary [l l] implies that simple COO-groups are 3CC-groups. The main purpose of this paper is to give a description of 3CC-groups. We proved:

On Homogeneous Standard Integral Table Algebras of Degree 4

ABSTRACT In this paper we classify, up to exact isomorphism, the algebras in the title which contain a faithful element and in which every non-trivial table subset has dimension at least five.