Classes of definite group matrices
Robert C. Thompson · Pacific Journal of Mathematics · 1966
Two positive definite symmetric n x n matrices A, B with integer elements and determinant one are said to be congruent if there exists an integral C such that B = CAC T {C τ is the transpose of C).This is an equivalence relation.The number of equivalence classes, C-classes, is finite and is known for all n ^ 16.Let G be a finite group of order n and let Y, Z be two positive definite symmetric group matrices for G with integral elements and determinant one.If an integral group matrix X for G exists such that Z = XYX T then Z, Y are said to be G-congruent.G congruence is an equivalence relation.In this paper the interlinking of the G-classes with the C-classes is determined for all groups of order n ^ 13.The principal result is that the G-class number is two for certain groups of orders eight or twelve and is one for all other groups of order n ^ 13.