Abel And Nilpotent Table Algebra Results

Arad and Blau abstracted the concept of table algebras, which are a class of finite dimensional, associative and commutative algebras over the complex numbers with certain specified properties, from the decompositions of products of either irreducible characters or conjugacy classes of finite groups. They provide a method in a uniform way to research the two cases mentioned above and have been a relatively independent researching subject.The present paper, from the point of view of the relationship between substructure and quotient structure, mainly characterizes structures of abelian and nilpotent table algebras and gains some meaningful results.Arad and Blau proved that an abelian table algebra (A, B) can be viewed as a group algebra of some abelian group G. Chapter 3 of this paper gives the structural theorem of abelian table algebras by defining a group structure in table basis. Furthermore, the structure of elementary abelian table algebras is discussed using the number of composition series of table algebras.Chapter 4 mainly deals with the nilpotency of table algebras. As is known, groups satisfying central extension conditions are nilpotent. we, firstly, obtain some basic properties on nilpotent table algebras, and then prove that table algebras which satisfy nilpotent extension are nilpotent by using Jordan-Holder type theorem of table algebras, i.e. nilpotency of a table algebra depends on that of their table subsets and quotient subsets, which is not fully corresponding to extension problems of nilpotent groups. Furthermore, we will find that conditions in the above theorem are not valid for abelian table algebras.