On the table of marks of a direct product of finite groups

We present a method for computing the table of marks of a direct product of finite groups. In contrast to the character table of a direct product of two finite groups, its table of marks is not simply the Kronecker product of the tables of marks of the two groups. Based on a decomposition of the inclusion order on the subgroup lattice of a direct product as a relation product of three smaller partial orders, we describe the table of marks of the direct product essentially as a matrix product of three class incidence matrices. Each of these matrices is in turn described as a sparse block diagonal matrix. As an application, we use a variant of this matrix product to construct a ghost ring and a mark homomorphism for the rational double Burnside algebra of the symmetric group S_3

Computing with finite groups

The character table of a finite group G is constructed by computing the eigenvectors of matrix equations determined by the centre of the group algebra. The numerical character values are expressed in algebraic form. A variant using a certain sub-algebra of the centre of the group algebra is used to ease problems associated with determining the conjugacy classes of elements of G. The simple group of order 50,232,960 and its subgroups PSL(2,17) and PSL(2,19) are constructed using general techniques. A combination of hand and machine calculation gives the character tables of the known simple groups of order < 106 excepting Sp(4,4) and PSL(2,q). The characters of the non- Abelian 2-groups of order < 2 6 are computed. Miscellaneous computations involving the symmetric group Sn are given

On the table of marks of a direct product of finite groups

We present a method for computing the table of marks of a direct product of finite groups. In contrast to the character table of a direct product of two finite groups, its table of marks is not simply the Kronecker product of the tables of marks of the two groups. Based on a decomposition of the inclusion order on the subgroup lattice of a direct product as a relation product of three smaller partial orders, we describe the table of marks of the direct product essentially as a matrix product of three class incidence matrices. Each of these matrices is in turn described as a sparse block diagonal matrix. As an application, we use a variant of this matrix product to construct a ghost ring and a mark homomorphism for the rational double Burnside algebra of the symmetric group S_3

Computing generators of the unit group of an integral abelian group ring

We describe an algorithm for obtaining generators of the unit group of the integral group ring ZG of a finite abelian group G. We used our implementation in Magma of this algorithm to compute the unit groups of ZG for G of order up to 110. In particular for those cases we obtained the index of the group of Hoechsmann units in the full unit group. At the end of the paper we describe an algorithm for the more general problem of finding generators of an arithmetic group corresponding to a diagonalizable algebraic group

Reliability and reproducibility of Atlas information

We discuss the reliability and reproducibility of much of the information contained in the Atlas of Finite Groups