Group-Pattern Matrices

A group-pattern for a group G of order n is specified by the n x n interior of any multiplication table for G in which the identity element e of G occupies each of the n principal diagonal positions. Thus, G and a list of the elements in G provide a pattern for the components of corresponding group-pattern matrices. As an example, there is at least one list of the elements in a cyclic group of order n for which the corresponding multiplication table specifies the pattern for n x n circulant matrices. This monograph shows that a serious treatment of its subject immediately yields an efficient Mathematica program to obtain all of the automorphisms for finite groups. This viewpoint is quite different from the way that Georg Frobenius introduced "group-matrices" for his research of 1896-1897. That distinction is maintained with the more descriptive phrase "group-pattern matrices" for the perspective here. Chapters may be read in various orders. As an illustration, to supplement an interest in circulant matrices, a reader may prefer to begin directly with Appendices A, B, C, D followed by Chapters 5, 6, 1, and 2. Or, for those whose primary interest may be automorphisms for finite groups, the recommended order would be Chapters 1, 2, 3, Appendix F, and Chapters 18, 19. Chapter 1 provides numerous group-patterns based on multiplication tables where the operations are easy to interpret and apply. Chapter 2 includes an algorithm to decide whether a given n x n matrix having n distinct components in its first row is a group-pattern matrix. An efficient Mathematica program implements that algorithm. When the decision is yes, the program specifies a suitable group and group-pattern. Chapter 3 presents on page 34 the computer-algebra program to obtain all of the automorphisms for finite groups. Chapter 4 is devoted to general algebraic properties of group-pattern matrices that have components in a field. Henceforth, for simplicity of explanation, let all components of matrices be complex numbers. Chapter 5 shows that, at one time, there was considerable interest in various determinants having interesting factorizations. Now, when those determinants are recognized as determinants of group-pattern matrices for abelian groups, their factorizations can be obtained as applications of a general result mentioned next. Chapters 6 through 9 are about group-pattern matrices for which the group is abelian. In particular, each such matrix is diagonalizable and its determinant is therefore equal to the product of the corresponding diagonal components. Moreover, when a group-pattern is specified for an abelian group G of order n, then: (i) there is a nonsingular n x n matrix that diagonalizes each n x n matrix having that pattern; (ii) the set of matrices having that pattern forms a ring; and (iii) that ring is isomorphic to the ring of n x n diagonal matrices. Chapters 10 and 11 show how research of Richard Dedekind in 1886 influenced Georg Frobenius to construct in 1897 a remarkable block-diagonalization for a group-pattern matrix where the group is the nonabelian one of order 6. That example motivated deep research about matrix representations of finite groups. Chapters 12 through 16 show how matrix representations for groups enable a Frobenius block-diagonalization to be obtained for any group-pattern matrix. Each such block-diagonalization is a diagonalization if and only if the group is abelian. There are 23 chapters and 8 appendices. Throughout, various computer-algebra programs are employed. For each one, there is a corresponding Mathematica notebook available that can be downloaded according to directions in Appendix H on page 235.