Generalizing Some Results In Group Theory To Table Algebra

In this paper, we discuss chiefly a class of particular algebra:table algebra which is an algebra having a distinguished basis.In particular, if the basis of a table algebra forms a group, we call it a group-like table algebra.Obviously, this kind of table algebras have the same properties as groups.But for the general table algebra, its basis also has similar structures and properties as a group.In view of this, we generalize some classic results in group theory to table algebra:let B=(UlsCbiC the disjoint union of double cosets of B, where C is a closed set of B and bi∈L(B), i=1,2,3,..., s, then there exists a subset of B such that it is not only a right coset representatives of C in B but also a left coset representatives; The distinguished basis B of a table algebra (A, B) cannot be the union of two proper closed subsets but it may be the union of three proper closed subsets;If there is normal closed subset C of B such that the quotient B’//C’ is isomorphic to Klein four group K4, then B can be written as the union of three proper closed subsets.In addition, by calculating we found that if B can be written as the union of three proper closed subsets then there exists a normal closed subset C of B such that the cardinality of B’//C’ is4for the table algebra whose dimension is4,5, or6. Finally, by giving counter-examples, we explain there is not only two cases such that T is tight in example2.21.